🧠:[<think>Okay, so I have this problem here: Given z equals (1 minus i) divided by (2 plus 2i), and I need to find the complex conjugate of z, which is denoted as (bar{z}). The options are A through D, with some involving fractions and i. Let me think step by step about how to approach this.First, I remember that the complex conjugate of a complex number is found by changing the sign of the imaginary part. For example, if you have a complex number a + bi, its conjugate is a - bi. But here, z is given as a fraction, so I might need to simplify z first before finding its conjugate. Alternatively, maybe I can find the conjugate directly by taking the conjugate of the numerator and denominator separately? Wait, let me recall: when you take the conjugate of a quotient, it's the same as the quotient of the conjugates. So, (overline{left(frac{a}{b}right)} = frac{overline{a}}{overline{b}}). So perhaps I can apply that property here.So, z is (1 - i)/(2 + 2i). Therefore, the conjugate of z should be the conjugate of the numerator over the conjugate of the denominator. The numerator is 1 - i, so its conjugate is 1 + i. The denominator is 2 + 2i, so its conjugate is 2 - 2i. Therefore, (bar{z} = frac{1 + i}{2 - 2i}). Then I might need to simplify that expression.Alternatively, maybe I should first simplify z by rationalizing the denominator, and then take the conjugate? Let me try both approaches and see which one is easier.Starting with the original z: (1 - i)/(2 + 2i). To simplify this, I can multiply numerator and denominator by the conjugate of the denominator, which is 2 - 2i. That will rationalize the denominator.So:z = [(1 - i)/(2 + 2i)] * [(2 - 2i)/(2 - 2i)] = [(1 - i)(2 - 2i)] / [(2 + 2i)(2 - 2i)]Let me compute the denominator first. Multiplying (2 + 2i)(2 - 2i). That's a difference of squares, so (2)^2 - (2i)^2 = 4 - (4i^2) = 4 - 4(-1) = 4 + 4 = 8. So denominator is 8.Now the numerator: (1 - i)(2 - 2i). Let's expand this:1*2 + 1*(-2i) - i*2 + (-i)*(-2i)= 2 - 2i - 2i + 2i^2Combine like terms:2 - 4i + 2i^2But i^2 is -1, so 2i^2 is 2*(-1) = -2.So numerator becomes 2 - 4i - 2 = (2 - 2) - 4i = 0 - 4i = -4iTherefore, z = (-4i)/8 = -i/2So z simplifies to -i/2. Therefore, the conjugate of z, which is (bar{z}), would be the conjugate of -i/2. Since -i/2 is purely imaginary (no real part), its conjugate is just the negative of the imaginary part, so i/2. Therefore, (bar{z} = frac{1}{2}i), which is option D.Wait, let me check. Alternatively, if I take the conjugate first before simplifying, as I thought earlier. The conjugate of z is [conjugate of (1 - i)] / [conjugate of (2 + 2i)] = (1 + i)/(2 - 2i). Then, simplifying that:Multiply numerator and denominator by the conjugate of the denominator again, which is 2 + 2i:[(1 + i)(2 + 2i)] / [(2 - 2i)(2 + 2i)]Denominator is same as before: (2)^2 - (2i)^2 = 4 - (-4) = 8.Numerator: (1 + i)(2 + 2i) = 1*2 + 1*2i + i*2 + i*2i = 2 + 2i + 2i + 2i^2 = 2 + 4i + 2*(-1) = 2 + 4i - 2 = 4iTherefore, the conjugate is (4i)/8 = i/2. Same result. So that's option D. So whether I simplify first and then take the conjugate or take the conjugate first and then simplify, I end up with the same answer. So the answer should be D: (frac{1}{2}i).Alternatively, another way to check: since z is -i/2, then the conjugate of z is i/2. So that's straightforward. If a complex number is a purely imaginary number, then its conjugate is just the negative of the imaginary part. Wait, no. Wait, if z = a + bi, then conjugate is a - bi. So if z is 0 - (1/2)i, then conjugate is 0 + (1/2)i, which is (1/2)i. Correct. So yes, the answer is D.Let me just verify once again.Original z: (1 - i)/(2 + 2i)Multiply numerator and denominator by conjugate of denominator:[(1 - i)(2 - 2i)] / (8)Numerator:(1)(2) + (1)(-2i) + (-i)(2) + (-i)(-2i) = 2 - 2i - 2i + 2i^2= 2 - 4i + 2(-1) = 2 -4i -2 = -4iThus, z = -4i / 8 = -i/2Therefore, conjugate of z is i/2, which is D. So the answer is D. The options given include C: -1/2 i and D: 1/2 i. So D is correct. Therefore, the correct answer is D.Final Answerboxed{D}</think>Given ( z = frac{1 - i}{2 + 2i} ), we need to find the complex conjugate (bar{z}).First, we simplify ( z ) by rationalizing the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, ( 2 - 2i ):[z = frac{(1 - i)(2 - 2i)}{(2 + 2i)(2 - 2i)}]Calculating the denominator:[(2 + 2i)(2 - 2i) = 2^2 - (2i)^2 = 4 - 4(-1) = 4 + 4 = 8]Expanding the numerator:[(1 - i)(2 - 2i) = 1 cdot 2 + 1 cdot (-2i) - i cdot 2 + (-i) cdot (-2i) = 2 - 2i - 2i + 2i^2]Simplifying the numerator:[2 - 4i + 2(-1) = 2 - 4i - 2 = -4i]Thus, ( z ) simplifies to:[z = frac{-4i}{8} = -frac{i}{2}]The complex conjugate of ( z ), denoted (bar{z}), is the conjugate of (-frac{i}{2}). Since the complex conjugate of a purely imaginary number ( bi ) is ( -bi ), the conjugate of (-frac{i}{2}) is (frac{i}{2}).Therefore, the answer is (boxed{D}).,<think>Okay, so I need to find the complex conjugate of z, which is given by the fraction (1 - i) divided by (2 + 2i). Hmm, complex conjugates... right, the conjugate of a complex number a + bi is a - bi. But here, z is a fraction, so maybe I need to simplify z first before finding its conjugate? Or maybe I can find the conjugate directly by taking the conjugate of the numerator and denominator? Let me think.Wait, the complex conjugate of a quotient is the quotient of the conjugates. So, if z = numerator/denominator, then the conjugate of z is conjugate(numerator)/conjugate(denominator). That seems right. So, the conjugate of (1 - i) is (1 + i), and the conjugate of (2 + 2i) is (2 - 2i). Therefore, conjugate of z would be (1 + i)/(2 - 2i). Then maybe I can simplify that?Alternatively, maybe I should first simplify z by rationalizing the denominator. Let me try that approach. Let's start with the original z:z = (1 - i)/(2 + 2i)To rationalize the denominator, I can multiply numerator and denominator by the conjugate of the denominator, which is (2 - 2i):z = [(1 - i)(2 - 2i)] / [(2 + 2i)(2 - 2i)]Let me compute the denominator first. Multiplying (2 + 2i)(2 - 2i) is a difference of squares, so it should be (2)^2 - (2i)^2 = 4 - 4i^2. Since i^2 is -1, that becomes 4 - 4(-1) = 4 + 4 = 8. So denominator is 8.Now the numerator: (1 - i)(2 - 2i). Let's expand that:1*2 + 1*(-2i) - i*2 + i*2iWhich is 2 - 2i - 2i + 2i^2Combine like terms: 2 - 4i + 2i^2Again, i^2 is -1, so 2 - 4i + 2*(-1) = 2 - 4i - 2 = 0 - 4i = -4iSo numerator is -4i, denominator is 8, so z simplifies to (-4i)/8 = -i/2. So z = -i/2. Therefore, the conjugate of z, which is bar{z}, should be the conjugate of -i/2.What's the conjugate of -i/2? The conjugate of a purely imaginary number is its negative. So the conjugate of -i is i, so the conjugate of -i/2 is i/2. So bar{z} = i/2, which is option D.Wait, let me check my steps again to be sure. Starting with z = (1 - i)/(2 + 2i). Multiplying numerator and denominator by (2 - 2i):Numerator: (1 - i)(2 - 2i) = 1*2 + 1*(-2i) - i*2 + i*2i = 2 - 2i - 2i + 2i². Then, 2i² is 2*(-1) = -2. So 2 - 2i -2i -2 = (2 - 2) + (-2i -2i) = 0 -4i = -4i. Denominator: (2 + 2i)(2 - 2i) = 4 - (2i)^2 = 4 - (-4) = 8. So z = -4i/8 = -i/2. Therefore, conjugate is i/2. So the answer is D.Alternatively, if I had taken the conjugate first, as in bar{z} = conjugate((1 - i)/(2 + 2i)) = conjugate(1 - i)/conjugate(2 + 2i) = (1 + i)/(2 - 2i). Then, simplifying that:Multiply numerator and denominator by (2 + 2i):[(1 + i)(2 + 2i)] / [(2 - 2i)(2 + 2i)]Denominator is same as before: 8Numerator: 1*2 + 1*2i + i*2 + i*2i = 2 + 2i + 2i + 2i² = 2 + 4i + 2*(-1) = 2 +4i -2 = 4i. So numerator is 4i, denominator is 8, so bar{z} = 4i/8 = i/2. Same result. So yes, D is correct.Therefore, the answer should be D: (1/2)i.Final Answerboxed{D}</think>To find the complex conjugate (bar{z}) of (z = frac{1-i}{2+2i}), we start by rationalizing the denominator of (z):[z = frac{1-i}{2+2i} times frac{2-2i}{2-2i} = frac{(1-i)(2-2i)}{(2+2i)(2-2i)}]First, calculate the denominator:[(2+2i)(2-2i) = 2^2 - (2i)^2 = 4 - 4(-1) = 4 + 4 = 8]Next, calculate the numerator:[(1-i)(2-2i) = 1 cdot 2 + 1 cdot (-2i) - i cdot 2 + i cdot (-2i) = 2 - 2i - 2i + 2i^2 = 2 - 4i + 2(-1) = 2 - 4i - 2 = -4i]Thus, the simplified form of (z) is:[z = frac{-4i}{8} = -frac{i}{2}]The complex conjugate (bar{z}) of (z = -frac{i}{2}) is obtained by changing the sign of the imaginary part:[bar{z} = frac{i}{2}]Therefore, the correct answer is (boxed{D}).]

🔑:To solve for bar{z} given z=frac{1-i}{2+2i}, we first simplify z as follows:[z = frac{1-i}{2+2i} = frac{1-i}{2(1+i)}]To simplify the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:[= frac{1-i}{2} cdot frac{1-i}{1+i}]Expanding the numerator and using the fact that i^2 = -1:[= frac{1 - 2i + i^2}{2(1+i)(1-i)}][= frac{1 - 2i - 1}{2(1 - i^2)}][= frac{-2i}{2(2)}][= -frac{i}{2}]Therefore, the conjugate of z, denoted as bar{z}, is found by changing the sign of the imaginary part:[bar{z} = frac{1}{2}i]Thus, the correct answer is boxed{D}.

🧠:[<think>Okay, let me try to figure out this problem step by step. So, we have two sets, A and B, and we need to find their intersection, A ∩ B. The answer choices are given, so maybe after solving both sets, I can compare them and see which interval overlaps.First, let's tackle set A. The problem says A is defined as {x | 1 < 2^x < 8}. That means all real numbers x such that when you raise 2 to the power of x, the result is between 1 and 8. Hmm, okay. I remember that exponential functions like 2^x are increasing functions. So, as x increases, 2^x increases. That should help in finding the interval.So, let's break down the inequality: 1 < 2^x < 8. To solve for x, I can take the logarithm base 2 of all parts. Since logarithm is a monotonic function (it preserves the inequality direction because the base is greater than 1), that should work.Taking log2 of 1 is 0, because 2^0 = 1. Then, log2 of 2^x is just x. And log2 of 8 is 3, because 2^3 = 8. So, applying log2 to all parts:log2(1) < log2(2^x) < log2(8)Which simplifies to:0 < x < 3So, set A is the interval (0, 3). Got that.Now, moving on to set B. The problem states B = {x | y = sqrt(-x^2 - 2x + 8)}. So, this is the set of all real numbers x for which the expression under the square root is non-negative. Because the square root function is only defined for non-negative numbers in real numbers. Therefore, the condition here is that -x^2 - 2x + 8 ≥ 0.Let me write that inequality down:-x² - 2x + 8 ≥ 0Hmm, maybe I can multiply both sides by -1 to make it easier, but I have to remember that multiplying by a negative number reverses the inequality sign. Let me try that.Multiplying both sides by -1 gives:x² + 2x - 8 ≤ 0Okay, so now we have a quadratic inequality: x² + 2x - 8 ≤ 0To solve this, first, let's find the roots of the quadratic equation x² + 2x - 8 = 0.Using the quadratic formula:x = [-b ± sqrt(b² - 4ac)] / 2aHere, a = 1, b = 2, c = -8.So,x = [-2 ± sqrt((2)^2 - 4*1*(-8))]/(2*1)Calculating discriminant:sqrt(4 + 32) = sqrt(36) = 6Thus, the roots are:x = [-2 + 6]/2 = 4/2 = 2x = [-2 - 6]/2 = -8/2 = -4So the roots are x = 2 and x = -4.Since the quadratic coefficient (a = 1) is positive, the parabola opens upwards. Therefore, the quadratic expression x² + 2x - 8 is ≤ 0 between its roots. So, the solution to x² + 2x - 8 ≤ 0 is x ∈ [-4, 2].Therefore, set B is the interval [-4, 2].Wait, let me double-check that. Because we had the original inequality -x² - 2x + 8 ≥ 0, which simplifies to x² + 2x - 8 ≤ 0. The solutions are where the parabola is below or equal to zero, which is between the roots. So yes, x from -4 to 2. So set B is [-4, 2].Alright, so now set A is (0, 3) and set B is [-4, 2]. The intersection of these two sets would be the overlap between (0, 3) and [-4, 2].Visualizing the intervals on the number line:Set A: starts at 0 (open circle) and goes up to 3 (open circle).Set B: starts at -4 (closed circle) and goes up to 2 (closed circle).The overlap between these two intervals would be from 0 (since A starts there) up to 2 (where B ends). However, since A is open at 0, the intersection will be open at 0. At 2, since B includes 2 but A is open at 3, but 2 is within A's interval (0,3). Wait, but A is (0,3), which is all numbers greater than 0 and less than 3. Set B is [-4, 2], which includes all numbers from -4 up to and including 2. Therefore, the intersection would be the set of numbers that are greater than 0 and less than or equal to 2. So, in interval notation, that's (0, 2].But let me verify that. The overlap is where both sets are true. So for x to be in both A and B, x must satisfy 0 < x < 3 and -4 ≤ x ≤ 2. Therefore, combining these, x must be greater than 0 and less than or equal to 2. Hence, (0, 2].Looking at the answer choices:A: [-4,3]B: (0,2]C: [-4,0)D: [2,3)So, the correct answer is option B: (0,2].But wait, let me check again. Because maybe I made a mistake with the endpoints.Set A is (0, 3). So it doesn't include 0 or 3. Set B is [-4, 2], which includes -4 and 2. The overlap is the set of x that are in both intervals. So x must be greater than 0 (since A starts at 0, not including it) and less than or equal to 2 (since B ends at 2, including it). Therefore, the intersection is (0, 2], which is option B. That seems right.But let me cross-verify with the answer choices. Option B is (0,2], which matches. The other options don't make sense. Option A is too broad, covering from -4 to 3, but A is only up to 3 and starts at 0. Option C is from -4 to 0, which doesn't overlap with A, which starts at 0. Option D is [2,3), which would be numbers from 2 to 3, but B only goes up to 2, so there's no overlap beyond 2. Therefore, D is invalid. So yes, B is correct.I think that's the right approach. Let me recap:1. Solve for set A: 1 < 2^x < 8. Convert to logarithmic inequalities: 0 < x < 3. So A is (0,3).2. Solve for set B: The expression under the square root must be non-negative. So -x^2 -2x +8 ≥0. Solving this quadratic inequality gives x between -4 and 2, inclusive. So B is [-4,2].3. Find the intersection of (0,3) and [-4,2]. The overlapping region is (0,2].4. Compare with answer choices. B is (0,2], so that's the answer.I don't see any mistakes in the reasoning. So I think the answer is B.Final Answerboxed{B}</think>To find ( A cap B ), we need to determine the intervals for sets ( A ) and ( B ) and then find their intersection.1. Set ( A ): - Given ( A = { x mid 1 < 2^x < 8 } ). - Solving the inequality ( 1 < 2^x < 8 ): - Taking the logarithm base 2 of all parts: ( log_2(1) < log_2(2^x) < log_2(8) ). - This simplifies to ( 0 < x < 3 ). - Therefore, ( A = (0, 3) ).2. Set ( B ): - Given ( B = { x mid y = sqrt{-x^2 - 2x + 8} } ). - The expression under the square root must be non-negative: ( -x^2 - 2x + 8 geq 0 ). - Solving the quadratic inequality ( -x^2 - 2x + 8 geq 0 ): - Multiplying by -1 (reversing the inequality): ( x^2 + 2x - 8 leq 0 ). - Finding the roots using the quadratic formula: ( x = frac{-2 pm sqrt{4 + 32}}{2} = frac{-2 pm 6}{2} ), which gives ( x = 2 ) and ( x = -4 ). - The quadratic expression ( x^2 + 2x - 8 leq 0 ) is satisfied between the roots: ( -4 leq x leq 2 ). - Therefore, ( B = [-4, 2] ).3. Finding ( A cap B ): - The interval for ( A ) is ( (0, 3) ). - The interval for ( B ) is ( [-4, 2] ). - The intersection of these intervals is the overlap between ( (0, 3) ) and ( [-4, 2] ), which is ( (0, 2] ).Thus, the final answer is (boxed{B}).,<think>Okay, so I need to find the intersection of sets A and B. Let me first understand what each set represents.Starting with set A: It's given as {x | 1 < 2^x < 8}. Hmm, that means all real numbers x where 2 raised to the power of x is between 1 and 8. Let me solve this inequality step by step.First, 1 < 2^x < 8. Since 2^x is a continuous and increasing function, I can solve the inequalities separately.For the left part: 1 < 2^x. Taking logarithm base 2 on both sides, log2(1) < x. Since log2(1) is 0, this simplifies to x > 0.For the right part: 2^x < 8. Similarly, take log2 of both sides: x < log2(8). Since 8 is 2^3, log2(8) is 3. So x < 3.Therefore, combining both inequalities, set A is (0, 3). So all real numbers x where x is greater than 0 and less than 3.Now moving on to set B: {x | y = sqrt(-x^2 -2x +8)}. Hmm, this is the set of all x such that the expression under the square root is non-negative because you can't take the square root of a negative number. So the domain here requires that -x^2 -2x +8 >= 0.Let me write that inequality: -x² -2x +8 ≥ 0. Maybe I can multiply both sides by -1 to make it easier, but I have to reverse the inequality sign when multiplying by a negative. Let's see:Multiplying by -1 gives: x² + 2x -8 ≤ 0.Now I need to solve the quadratic inequality x² + 2x -8 ≤ 0.First, find the roots of the equation x² + 2x -8 = 0. Using the quadratic formula:x = [-b ± sqrt(b² - 4ac)] / 2aHere, a = 1, b = 2, c = -8.So discriminant D = (2)^2 - 4*1*(-8) = 4 + 32 = 36.Square root of 36 is 6. Therefore,x = [-2 ± 6]/2.First root: (-2 + 6)/2 = 4/2 = 2.Second root: (-2 -6)/2 = -8/2 = -4.So the quadratic equation x² + 2x -8 = 0 has roots at x = -4 and x = 2.Since the quadratic opens upwards (coefficient of x² is positive), the quadratic is ≤ 0 between the roots. So the solution to x² + 2x -8 ≤ 0 is x ∈ [-4, 2]. Therefore, the original inequality -x² -2x +8 ≥ 0 is also satisfied in [-4, 2]. Therefore, set B is [-4, 2].So now we have:A = (0, 3)B = [-4, 2]The intersection of A and B would be the set of all x that are in both A and B. So, overlapping region between (0,3) and [-4,2]. Let's visualize this.The interval (0,3) is from 0 to 3, not including the endpoints. The interval [-4,2] is from -4 to 2, including both endpoints. The overlap between these two intervals would be from 0 to 2. Since in A, 0 is not included, and in B, 2 is included. Therefore, the intersection is (0, 2].Wait, but let me verify again. The intersection of (0,3) and [-4,2] is all x such that x >0 and x <3, and also x >= -4 and x <=2. So combining these, the x must satisfy both x >0 and x <=2. So that's (0, 2].Now looking at the answer choices:A: [-4, 3]B: (0, 2]C: [-4, 0)D: [2, 3)So the correct answer should be B: (0, 2]. Let me cross-check.For set A, x must be between 0 and 3, not including 0 and 3. For set B, x must be between -4 and 2, including both. So the numbers that are in both sets must be greater than 0 (since A starts at 0 exclusive) and up to 2 (since B ends at 2 inclusive). Therefore, the intersection is (0, 2], which is option B.Wait, but let me check if there's a possibility that maybe I made a mistake in solving set B. Let me re-examine set B.Set B is defined as {x | y = sqrt(-x² -2x +8)}. The domain requires that -x² -2x +8 >= 0. So, solving that:-x² -2x +8 >=0Multiply both sides by -1 (reversing inequality):x² + 2x -8 <=0Which factors as (x + 4)(x - 2) <=0. So critical points at x=-4 and x=2. Testing intervals:For x < -4, say x=-5: (-5 +4)(-5 -2) = (-1)(-7) = 7 >0.For x between -4 and 2, say x=0: (0 +4)(0 -2)=4*(-2)=-8 <0.For x >2, say x=3: (3+4)(3-2)=7*1=7>0.So the inequality x² +2x -8 <=0 holds when x is between -4 and 2, inclusive. Therefore, set B is [-4, 2], which matches my previous result.Therefore, intersection with A (0,3) is indeed (0,2], which is answer B.But let me check the options again. The options are A: [-4,3], which is too broad. B: (0,2], which is what we found. C: [-4,0), which is the left part, but A doesn't include anything less than 0. D: [2,3), but B ends at 2, so the intersection beyond 2 is not possible. Hence, B is correct.Final Answerboxed{B}</think>To find the intersection of sets (A) and (B):1. Set (A): Given by ( { x mid 1 < 2^x < 8 } ). - Solving (1 < 2^x < 8): - (1 < 2^x) implies (x > 0) (since (log_2 1 = 0)). - (2^x < 8) implies (x < 3) (since (log_2 8 = 3)). - Therefore, (A = (0, 3)).2. Set (B): Given by ( { x mid y = sqrt{-x^2 - 2x + 8} } ). - The expression under the square root must be non-negative: (-x^2 - 2x + 8 geq 0). - Solving (-x^2 - 2x + 8 geq 0): - Multiply by (-1) (reversing the inequality): (x^2 + 2x - 8 leq 0). - Solving the quadratic inequality (x^2 + 2x - 8 leq 0) using the quadratic formula: - Roots are (x = -4) and (x = 2). - The quadratic is non-positive between the roots: ([-4, 2]). - Therefore, (B = [-4, 2]).3. Intersection (A cap B): - (A = (0, 3)) - (B = [-4, 2]) - The intersection is the overlap of these intervals: ((0, 2]).The final answer is (boxed{B}).]

🔑:To find A cap B, we need to determine the sets A and B first and then find their intersection.Step 1: Determine Set AGiven A = {x | 1 1Taking the logarithm base 2 on both sides:log_2(2^x) > log_2(1)x > 0For the upper bound, 2^x < 8:2^x < 2^3Since the base is the same, we can equate the exponents:x < 3Combining both inequalities, we get:0 < x < 3Step 2: Determine Set BGiven B = {x | y = sqrt{-x^2 - 2x + 8}}, for y to be real, the expression under the square root must be non-negative:-x^2 - 2x + 8 geqslant 0Rearranging, we get:x^2 + 2x - 8 leqslant 0To solve this quadratic inequality, we find the roots of the equation x^2 + 2x - 8 = 0:(x + 4)(x - 2) = 0So, the roots are x = -4 and x = 2.The inequality x^2 + 2x - 8 leqslant 0 holds between the roots, so:-4 leqslant x leqslant 2Step 3: Find A cap BNow, we have:- A = {x | 0 < x < 3}- B = {x | -4 leqslant x leqslant 2}The intersection A cap B is the set of x values that satisfy both conditions:A cap B = (0, 2]Therefore, the correct answer is boxed{B}.

🧠:[<think>Okay, let's try to figure out how to find the area of a trapezoid when we know the lengths of the diagonals and the midline. Hmm, the problem states that the diagonals are 6 and 8, and the midline is 5. We need to find the area. First, I remember that the area of a trapezoid is given by the formula: [text{Area} = text{midline} times text{height}]So, if we can find the height, we can multiply it by the midline (which is 5) to get the area. But how do we find the height when we only know the diagonals? Let me recall some properties of trapezoids. The midline is the average of the lengths of the two bases. So, if the midline is 5, then the sum of the two bases is 10. Let's denote the bases as ( a ) and ( b ), so:[frac{a + b}{2} = 5 implies a + b = 10]Got that. Now, the diagonals are 6 and 8. I need to relate the diagonals to the height. Hmm. I remember that in a trapezoid, the lengths of the diagonals can be related to the sides and the height through some formulas, but I'm not exactly sure. Maybe I need to use coordinate geometry here?Let me try setting up a coordinate system. Let's place the trapezoid so that the bases are horizontal. Let the lower base be ( a ) and the upper base be ( b ). Let me assign coordinates to the vertices. Let’s say the lower left vertex is at the origin (0, 0), and the lower right vertex is at (a, 0). Then, the upper base is of length ( b ), so the upper left vertex would be at (x, h) and the upper right vertex at (x + b, h), where ( h ) is the height. The midline is 5, so ( a + b = 10 ).Now, the diagonals are from (0, 0) to (x + b, h) and from (a, 0) to (x, h). Their lengths are 8 and 6, respectively. Wait, but which diagonal is which? The problem says the diagonals are 6 and 8. Maybe it doesn't specify which is which, so maybe I have to consider both possibilities. Hmm. Maybe it doesn't matter? Let's see.So, first diagonal: from (0, 0) to (x + b, h). The length is ( sqrt{(x + b)^2 + h^2} ). Second diagonal: from (a, 0) to (x, h). The length is ( sqrt{(a - x)^2 + h^2} ). But we also know that the upper base is of length ( b ), so the distance between (x, h) and (x + b, h) is ( b ). Similarly, the lower base is ( a ). The legs of the trapezoid are the non-parallel sides. The legs would be from (0, 0) to (x, h) and from (a, 0) to (x + b, h). Their lengths can be expressed as ( sqrt{x^2 + h^2} ) and ( sqrt{(a - x - b)^2 + h^2} ), respectively. But maybe we don't need the legs. Wait, but since the problem doesn't specify whether the trapezoid is isosceles or not, we have to consider a general trapezoid. This might get complicated with too many variables. Let's see: we have variables ( a ), ( b ), ( x ), ( h ). But we know that ( a + b = 10 ), so we can express ( a = 10 - b ). So, maybe we can reduce the number of variables. So, let's rewrite the coordinates. The lower base is ( a = 10 - b ), upper base is ( b ). Coordinates are: lower left (0, 0), lower right (10 - b, 0), upper left (x, h), upper right (x + b, h). Now, the two diagonals are from (0, 0) to (x + b, h) with length 8, and from (10 - b, 0) to (x, h) with length 6. So:First diagonal: [sqrt{(x + b)^2 + h^2} = 8 implies (x + b)^2 + h^2 = 64]Second diagonal: [sqrt{(10 - b - x)^2 + h^2} = 6 implies (10 - b - x)^2 + h^2 = 36]So now we have two equations:1. ( (x + b)^2 + h^2 = 64 )2. ( (10 - b - x)^2 + h^2 = 36 )If we subtract the second equation from the first, we can eliminate ( h^2 ):[(x + b)^2 - (10 - b - x)^2 = 64 - 36 = 28]Let me expand both squares:First term: ( (x + b)^2 = x^2 + 2bx + b^2 )Second term: ( (10 - b - x)^2 = (10 - b - x)^2 = (-x + (10 - b))^2 = x^2 - 2(10 - b)x + (10 - b)^2 )Subtracting the second from the first:[[x^2 + 2bx + b^2] - [x^2 - 2(10 - b)x + (10 - b)^2] = 28]Simplify:( x^2 + 2bx + b^2 - x^2 + 2(10 - b)x - (10 - b)^2 = 28 )Simplify term by term:- ( x^2 - x^2 = 0 )- ( 2bx + 2(10 - b)x = 2x[b + 10 - b] = 2x(10) = 20x )- ( b^2 - (10 - b)^2 = b^2 - [100 - 20b + b^2] = b^2 - 100 + 20b - b^2 = 20b - 100 )So combining:( 20x + 20b - 100 = 28 )Simplify:( 20x + 20b = 128 )Divide both sides by 20:( x + b = 6.4 )So ( x = 6.4 - b )Now, we can substitute this back into one of the original equations. Let's take the first equation:( (x + b)^2 + h^2 = 64 )But ( x + b = 6.4 ), so:( 6.4^2 + h^2 = 64 )Calculate 6.4 squared: 6.4 * 6.4. Let's compute that:6 * 6 = 36, 6 * 0.4 = 2.4, 0.4 * 6 = 2.4, 0.4 * 0.4 = 0.16. So total: 36 + 2.4 + 2.4 + 0.16 = 40.96So:( 40.96 + h^2 = 64 implies h^2 = 64 - 40.96 = 23.04 implies h = sqrt{23.04} )Calculate sqrt(23.04). Let me see:4.8 squared is 23.04 (since 5^2 = 25, 4.8^2 = (5 - 0.2)^2 = 25 - 2*5*0.2 + 0.2^2 = 25 - 2 + 0.04 = 23.04). So h = 4.8.So height is 4.8. Then the area is midline * height = 5 * 4.8 = 24. So the area is 24.Wait, but let me verify this because sometimes when you set up coordinates, you might have made an assumption that isn't valid.Wait, we found h = 4.8. Let me check with the second equation to make sure.We had:( (10 - b - x)^2 + h^2 = 36 )But since we found x = 6.4 - b, then 10 - b - x = 10 - b - (6.4 - b) = 10 - b -6.4 + b = 3.6So, (3.6)^2 + h^2 = 36?Calculate 3.6 squared: 12.96. Then 12.96 + h^2 = 36. Therefore, h^2 = 36 -12.96 = 23.04, which is the same as before. So h is indeed 4.8.Therefore, the area is midline * height = 5 * 4.8 = 24. So, 24. Hmm, seems straightforward. But let me check if there is another approach or if I missed something.Another way to think about it: There's a formula that relates the area of a trapezoid to its diagonals and the angle between them. But I don't remember it exactly. Let me recall.If two diagonals are d1 and d2, and the angle between them is θ, then the area is (d1*d2*sinθ)/2. But this is for a parallelogram. For a trapezoid, it's more complicated because the diagonals aren't necessarily intersecting at the same angle. Wait, maybe not. Hmm.Alternatively, perhaps using vectors. Suppose the diagonals are vectors. But maybe this is overcomplicating. Since we already got an answer through coordinate geometry, perhaps that's the way to go. But let me verify with another method.Alternatively, in some textbooks, there's a formula for the area of a trapezoid in terms of the diagonals and the midline. Let me see if I can derive that.Let’s denote the midline as m = 5, so a + b = 10. Let’s denote the diagonals as d1 = 6 and d2 = 8. Wait, but in the problem statement, it says the diagonals are 6 and 8. In my coordinate setup, one diagonal was 8 and the other was 6. But does it matter which one is which? Let's check.Suppose instead that the diagonal from (0,0) to (x + b, h) is 6 and the other is 8. Then, the same equations would be:First diagonal: ( (x + b)^2 + h^2 = 36 )Second diagonal: ( (10 - b - x)^2 + h^2 = 64 )Subtracting the first equation from the second:( (10 - b - x)^2 - (x + b)^2 = 64 - 36 = 28 )Expanding:Same as before, the left-hand side becomes 20x + 20b - 100. So:20x + 20b - 100 = 2820x + 20b = 128x + b = 6.4Same result as before. Then, substituting back:From first equation: (6.4)^2 + h^2 = 36. But 6.4 squared is 40.96, so h^2 = 36 - 40.96 = -4.96, which is impossible. So, this case is invalid. Therefore, the diagonal lengths must be 8 and 6 with the first diagonal (from lower left to upper right) being 8 and the other being 6. So, our initial assumption was correct. Therefore, h = 4.8 is valid.Therefore, the area is indeed 5 * 4.8 = 24.Alternatively, maybe there's a formula that if the diagonals are perpendicular, the area is (d1*d2)/2, but in this case, the diagonals are not necessarily perpendicular. Wait, but in general trapezoid, the area can be related to the product of the diagonals and the sine of the angle between them, but divided by 2 and multiplied by something else. Wait, maybe not. Let me think.If we consider the formula for the area of a quadrilateral: 1/2 * d1 * d2 * sinθ, where θ is the angle between the diagonals. But this formula is for a quadrilateral in general. However, in a trapezoid, the angle between the diagonals might be related to the height and the bases.But since we know the midline and the height, we can compute the area directly as midline * height. Since we found h = 4.8, then 5 * 4.8 = 24.Alternatively, perhaps there's a relation between the lengths of the diagonals, the midline, and the height. Let's see. Let's denote m = midline = 5, d1 = 8, d2 = 6. Then, is there a formula that connects m, d1, d2, and h?From our coordinate approach, we found that h = 4.8. Let's see if that can be derived in another way.Suppose we consider the two diagonals intersecting each other. Let’s denote the point of intersection as O. In a trapezoid, the diagonals divide each other proportionally. That is, the ratio of the segments of one diagonal is equal to the ratio of the segments of the other diagonal, and this ratio is equal to the ratio of the lengths of the two bases. However, since we know the midline is 5, then a + b = 10, so the ratio is a/b. Let’s denote the ratio as k = a/b. Then, the segments of the diagonals would be in the same ratio k.But maybe we can use this property. Let's denote that the diagonals intersect at point O, dividing each diagonal into segments. Let’s say diagonal d1 = 8 is divided into segments of length p and 8 - p, and diagonal d2 = 6 is divided into segments of length q and 6 - q. Then, the ratio p/(8 - p) = q/(6 - q) = a/b. But since a + b = 10, and a/b = k, then a = 10k/(k + 1), b = 10/(k + 1).But this might be a longer path. However, perhaps we can find the angle between the diagonals using the law of cosines in the triangles formed by the diagonals.Alternatively, consider the two triangles formed by the diagonals. In a trapezoid, the area is the sum of the areas of the two triangles formed by a diagonal. But since we have two different diagonals, maybe this is not straightforward.Wait, but in our coordinate system, we found the height directly. Maybe the coordinate approach was the most straightforward here. Since the problem didn't specify that the trapezoid is isosceles, we can't assume symmetry, so coordinate geometry seems necessary.Alternatively, another approach: in any trapezoid, the sum of the squares of the diagonals is equal to the sum of the squares of the non-parallel sides plus twice the product of the bases. Wait, no, that's in a parallelogram. Wait, in a parallelogram, the sum of the squares of the diagonals equals twice the sum of the squares of the sides. For a trapezoid, is there a similar formula?Let me check. In a trapezoid, there is a formula that relates the lengths of the diagonals to the lengths of the bases and the legs. Specifically, for a trapezoid with bases a and b, legs c and d, and diagonals p and q, the formula is:( p^2 + q^2 = c^2 + d^2 + 2ab )But I'm not sure if this is correct. Let me verify. Suppose we have a trapezoid. If we apply the law of cosines to the triangles formed by the diagonals.Wait, maybe in a trapezoid, the formula is similar to the parallelogram one but adjusted. Let me think. If we take the two diagonals, each diagonal divides the trapezoid into two triangles. Let’s consider one of the triangles. Suppose we take the triangle formed by one diagonal, then by applying the law of cosines to that triangle.But maybe this is getting too complicated. Alternatively, since we already found the height through coordinate geometry, perhaps that's the simplest way.Wait, but let me check if the answer makes sense. If the midline is 5, and the height is 4.8, then the area is 24. Let me see if there's another way to cross-validate this.Alternatively, think of the trapezoid as a combination of two triangles. But not sure. Alternatively, use vectors. Let me try.Let’s denote vectors for the diagonals. Let’s assume the trapezoid has vectors for the diagonals d1 and d2. Then, the area can be calculated as half the magnitude of the cross product of the diagonals. Wait, but that formula is for a parallelogram. For a general quadrilateral, the area is half the product of the lengths of the diagonals multiplied by the sine of the angle between them. But in a trapezoid, the angle between the diagonals is not necessarily the same as in a general quadrilateral. However, if we can find the angle between the diagonals, then we can compute the area as 1/2 * d1 * d2 * sin(theta).But how to find theta? Let's recall that in our coordinate system, we can find the angle between the diagonals. The diagonals are from (0,0) to (x + b, h) and from (10 - b, 0) to (x, h). Let's compute the vectors of these diagonals.First diagonal vector: (x + b, h)Second diagonal vector: (x - (10 - b), h - 0) = (x - 10 + b, h)The angle between them can be found using the dot product:cos(theta) = [ (x + b)(x - 10 + b) + h^2 ] / [ sqrt((x + b)^2 + h^2) * sqrt((x - 10 + b)^2 + h^2) ]But since we know sqrt((x + b)^2 + h^2) = 8 and sqrt((x - 10 + b)^2 + h^2) = 6, and also (x + b) = 6.4 from earlier, and (10 - b - x) = 3.6, which implies (x - 10 + b) = -3.6.So, substitute:cos(theta) = [ (6.4)(-3.6) + h^2 ] / [8 * 6]Compute numerator:6.4 * (-3.6) = -23.04h^2 = 23.04So total numerator: -23.04 + 23.04 = 0Therefore, cos(theta) = 0 / (8*6) = 0. Therefore, theta = 90 degrees. Wait, the angle between the diagonals is 90 degrees? So the diagonals are perpendicular?But if that's the case, then the area of the quadrilateral would be 1/2 * d1 * d2 = 1/2 * 8 * 6 = 24. Which matches the area we found earlier! So, that's another way to get the area. Therefore, since the angle between the diagonals is 90 degrees, the area is 24.But wait, how come we didn't use the midline here? Because in this approach, we didn't use the midline, but we got the same answer. However, the midline was necessary to determine that the angle between the diagonals is 90 degrees.So, in this case, the midline and the lengths of the diagonals allowed us to find that the angle between the diagonals is 90 degrees, hence the area is 1/2 * 6 * 8 = 24. But this is a special case where the diagonals are perpendicular. Therefore, if in a trapezoid the diagonals are perpendicular, then the area is half the product of the diagonals. However, in general, this is not true unless the diagonals are perpendicular. Here, we found that the angle between the diagonals is 90 degrees, so this formula applies. But how did we come to that?From the coordinate approach, we found that the numerator in the cosine formula was zero, leading to theta = 90 degrees. Therefore, the area can be computed as 1/2 * 6 * 8 = 24. But the midline being 5 was crucial in leading us to find that the angle is 90 degrees. Therefore, the answer is 24. So, either way, whether through coordinate geometry leading us to the height, or through recognizing that the diagonals are perpendicular and using the formula for the area in that case, we arrive at the same answer. Therefore, the area of the trapezoid is 24.Final AnswerThe area of the trapezoid is boxed{24}.</think>Given a trapezoid with diagonals of lengths 6 and 8, and a midline of 5, we need to find the area of the trapezoid.1. Midline and Bases: - The midline ( m ) of a trapezoid is the average of the lengths of the two bases ( a ) and ( b ). - Given the midline is 5, we have: [ frac{a + b}{2} = 5 implies a + b = 10 ]2. Coordinate Geometry Setup: - Place the trapezoid in a coordinate system with the lower base ( a ) and upper base ( b ). - The coordinates of the vertices are set such that the lower left vertex is at ( (0, 0) ), the lower right vertex at ( (a, 0) ), the upper left vertex at ( (x, h) ), and the upper right vertex at ( (x + b, h) ).3. Diagonal Lengths: - The lengths of the diagonals are given as 8 and 6. Using the coordinates, we set up the equations for the diagonals: - Diagonal from ( (0, 0) ) to ( (x + b, h) ) with length 8: [ sqrt{(x + b)^2 + h^2} = 8 implies (x + b)^2 + h^2 = 64 ] - Diagonal from ( (a, 0) ) to ( (x, h) ) with length 6: [ sqrt{(a - x)^2 + h^2} = 6 implies (a - x)^2 + h^2 = 36 ]4. Solving for Variables: - Subtract the second equation from the first to eliminate ( h^2 ): [ (x + b)^2 - (10 - b - x)^2 = 28 ] - Expanding and simplifying: [ 20x + 20b - 100 = 28 implies x + b = 6.4 ] - Substitute ( x + b = 6.4 ) back into the first equation: [ (6.4)^2 + h^2 = 64 implies h^2 = 23.04 implies h = 4.8 ]5. Area Calculation: - The height ( h ) is found to be 4.8. The area of the trapezoid is given by the product of the midline and the height: [ text{Area} = text{midline} times text{height} = 5 times 4.8 = 24 ]6. Verification with Diagonals: - The angle between the diagonals is found to be 90 degrees (perpendicular), leading to the area formula for a quadrilateral with perpendicular diagonals: [ text{Area} = frac{1}{2} times d_1 times d_2 = frac{1}{2} times 6 times 8 = 24 ]Thus, the area of the trapezoid is (boxed{24}).,<think>Okay, so I need to find the area of a trapezoid where the diagonals are 6 and 8, and the midline is 5. Hmm, let's recall some properties of trapezoids. The midline (or mid-segment) of a trapezoid is the segment that connects the midpoints of the non-parallel sides (the legs). The length of the midline is equal to the average of the lengths of the two bases. So if the bases are of lengths a and b, then the midline m is (a + b)/2. The problem says the midline is 5, so that means (a + b)/2 = 5, so a + b = 10. Got that.Now, the area of a trapezoid is given by the formula: Area = (1/2)*(a + b)*h, where h is the height. Since we already know that a + b = 10, the area simplifies to (1/2)*10*h = 5h. So if I can find the height h, I can compute the area. But how?The problem gives the lengths of the diagonals as 6 and 8. Hmm, how do the diagonals relate to the height or the bases? I remember that in some quadrilaterals like rectangles, the diagonals are equal, but trapezoids don't necessarily have equal diagonals. However, maybe there's a formula that relates the diagonals, the midline, and the area. Let me think.Alternatively, maybe I can use coordinate geometry. Let's place the trapezoid on a coordinate system to make it easier. Suppose the two bases are parallel to the x-axis. Let’s denote the lower base as AB with length a, and the upper base as CD with length b. The coordinates can be set such that point A is at (0, 0), point B is at (a, 0), point C is at (c, h), and point D is at (d, h). Since CD has length b, then |d - c| = b. Also, the midline is 5, so (a + b)/2 = 5, so a + b = 10.Now, the diagonals are AC and BD. The length of diagonal AC is 6, and BD is 8. Let's write expressions for these diagonals in terms of coordinates.First, diagonal AC connects A(0, 0) to C(c, h). The distance between A and C is sqrt[(c - 0)^2 + (h - 0)^2] = sqrt(c² + h²) = 6.Similarly, diagonal BD connects B(a, 0) to D(d, h). The distance between B and D is sqrt[(d - a)^2 + (h - 0)^2] = sqrt[(d - a)² + h²] = 8.Also, since CD is the upper base with length b, then d - c = b (assuming d > c). But since the trapezoid can be oriented either way, maybe the difference is |d - c| = b. For simplicity, let's assume d > c, so d = c + b.Additionally, since the trapezoid has sides AD and BC as the legs. If the trapezoid is not isosceles, the legs can be of different lengths. But since we don't know if it's isosceles, we can't assume that. However, maybe there is another relationship here.Wait, but perhaps we can relate coordinates with the midline. The midline connects the midpoints of the legs. The midpoint of AD is ((0 + d)/2, (0 + h)/2) = (d/2, h/2), and the midpoint of BC is ((a + c)/2, (0 + h)/2) = ((a + c)/2, h/2). The midline is the segment connecting these two midpoints, and its length is 5. So the distance between (d/2, h/2) and ((a + c)/2, h/2) is 5. Since the y-coordinates are the same, the distance is just the difference in x-coordinates: |(d/2) - ((a + c)/2)| = |(d - a - c)/2| = 5.But we know that d = c + b (from earlier), so substituting that in:|((c + b) - a - c)/2| = |(b - a)/2| = 5.Therefore, |b - a| = 10. But we also know that a + b = 10. So we have two equations:1) a + b = 102) |b - a| = 10Let me solve these. Let's drop the absolute value for a second. So, either b - a = 10 or a - b = 10. But since a and b are lengths, they must be positive. Let's check the first case: b - a = 10. Then from a + b = 10, substituting b = a + 10 into a + (a + 10) = 10, we get 2a + 10 = 10 → 2a = 0 → a = 0. But a base can't have length 0. So that's impossible.Second case: a - b = 10. Then from a + b = 10, substituting a = b + 10, we get (b + 10) + b = 10 → 2b + 10 = 10 → 2b = 0 → b = 0. Again, impossible. Wait, this is a problem. Both cases lead to one of the bases having length 0, which is not possible. So maybe my assumption is wrong.But the midline is 5, so (a + b)/2 = 5, and |b - a| = 10. But this leads to a contradiction. That can't be. So either my coordinate setup is wrong, or my reasoning here is flawed.Wait, perhaps the midline length is 5, which gives (a + b)/2 = 5, so a + b = 10. Then the distance between the midpoints of the legs is 5. But in my coordinate system, I computed that |(d - a - c)/2| = 5. But maybe I messed up the coordinates. Let me check again.Coordinates: A(0,0), B(a,0), C(c,h), D(d,h). Then midpoints of AD is (d/2, h/2), midpoint of BC is ((a + c)/2, h/2). The distance between these two midpoints is |d/2 - (a + c)/2| = |(d - a - c)/2| = 5. So |d - a - c| = 10. But we know that d = c + b, so substituting:|c + b - a - c| = |b - a| = 10. So |b - a| = 10. But since a + b = 10, this would mean that either b = a + 10, leading to a + (a + 10) = 10 ⇒ 2a = 0 ⇒ a = 0, or a = b + 10, leading to (b + 10) + b = 10 ⇒ 2b = 0 ⇒ b = 0. Both impossible. Therefore, this suggests that my coordinate system might be set up incorrectly, or there's an error in my reasoning.Wait, maybe the midline is not connecting midpoints of the legs, but of the bases? Wait no, midline of a trapezoid is specifically the segment connecting midpoints of the legs, and its length is the average of the two bases. So the formula (a + b)/2 is correct. Therefore, if the midline is 5, a + b = 10. But how does this lead to |b - a| = 10? That can't be possible. Therefore, perhaps there's a mistake in the coordinate system approach here.Alternatively, maybe the problem is that the trapezoid is not convex? Or maybe my coordinate system assumes something about the trapezoid that isn't necessarily true. Let me think.Alternatively, perhaps the formula relating the midline and the diagonals. Wait, I recall that in a trapezoid, the sum of the squares of the diagonals is equal to the sum of the squares of the non-parallel sides plus 2 times the product of the bases. Wait, no, that might be for parallelograms. Let me check.In a parallelogram, the sum of the squares of the diagonals equals twice the sum of the squares of the sides. But in a trapezoid, there's a formula that relates the diagonals to the bases, legs, and the height. Maybe I need to derive such a formula.Alternatively, perhaps using vectors or trigonometry. Hmm.Alternatively, consider that the area can also be found if we know the angle between the diagonals. Wait, if we can find the angle between the diagonals, then the area can be computed as (1/2)*d1*d2*sin(theta), where theta is the angle between the diagonals. But is that applicable here? Wait, that formula is for the area of a quadrilateral when you know the lengths of the diagonals and the angle between them. However, in general quadrilaterals, the area is (1/2)*d1*d2*sin(theta)*sin(phi), where theta and phi are the angles between the diagonals? Wait, no, perhaps I need to recall the formula.Actually, for any quadrilateral, the area is (1/2)*d1*d2*sin(theta), where theta is the angle between the diagonals. But I might be misremembering. Wait, let's check.Suppose the two diagonals intersect at an angle theta. Then the area of the quadrilateral is (1/2)*d1*d2*sin(theta). Yes, that seems correct. Because each diagonal is divided into two parts by their intersection, and the area would be the sum of the areas of the four triangles, but maybe that formula is only valid for certain quadrilaterals. Wait, for a convex quadrilateral, if the diagonals intersect at angle theta, then the area is indeed (1/2)*d1*d2*sin(theta). Let me confirm.Yes, according to some references, the area of a convex quadrilateral with diagonals d1 and d2 intersecting at an angle theta is (1/2)*d1*d2*sin(theta). So if I can find theta, the angle between the diagonals, then I can compute the area.But how can I find theta? Well, maybe by relating it to the midline and the bases. Hmm.Wait, but I know the midline is 5, which gives me a + b = 10. The diagonals are 6 and 8. Maybe there's a formula that relates the diagonals, the midline, and the angle between the diagonals.Alternatively, let's consider the relationship between the diagonals and the midline. Let me try to think.Suppose in the trapezoid, the two diagonals are d1 = 6 and d2 = 8. Let’s denote the angle between them as theta. Then the area is (1/2)*6*8*sin(theta) = 24*sin(theta). But we also know that the area is 5h. So 24*sin(theta) = 5h. If I can relate h and theta, maybe through the midline.Alternatively, maybe using the law of cosines on the triangles formed by the diagonals. Let me try.When the diagonals intersect, they divide each other into segments. Let’s denote the point of intersection as O. Let’s say diagonal AC (length 6) is divided into segments AO and OC, and diagonal BD (length 8) is divided into segments BO and OD. In a trapezoid, the ratio of the segments of the diagonals is equal to the ratio of the lengths of the bases. That is, AO/OC = AB/CD = a/b. Similarly, BO/OD = a/b. This is a property of trapezoids.So, if AO/OC = a/b, then AO = (a/(a + b)) * AC = (a/10)*6 = (3a)/5. Similarly, OC = (b/10)*6 = (3b)/5. Similarly, BO = (a/10)*8 = (4a)/5, and OD = (b/10)*8 = (4b)/5.Now, in triangle AOB, we have sides AO = 3a/5, BO = 4a/5, and AB = a. Similarly, in triangle COD, we have sides CO = 3b/5, OD = 4b/5, and CD = b. Wait, but actually, triangles AOB and COD are similar to the original triangles formed by the bases and the legs? Not sure. Alternatively, maybe we can use the law of cosines on the triangles formed by the diagonals.Consider triangle AOB: sides AO = 3a/5, BO = 4a/5, and AB = a. Wait, but 3a/5 + 4a/5 = 7a/5, which is greater than a only if 7a/5 > a, which is true since 7/5 > 1. So the triangle inequality holds. Wait, but actually, AO and BO are parts of the diagonals, so maybe the lengths are 3a/5 and 4a/5. Wait, no, hold on. If AO = (a/(a + b)) * AC = (a/10)*6 = 3a/5, similarly BO = (a/(a + b)) * BD = (a/10)*8 = 4a/5. So in triangle AOB, sides AO = 3a/5, BO = 4a/5, and AB = a. Then using the law of cosines on triangle AOB:AB² = AO² + BO² - 2*AO*BO*cos(theta1), where theta1 is the angle between AO and BO.So, a² = (9a²/25) + (16a²/25) - 2*(3a/5)*(4a/5)*cos(theta1)Simplify:a² = (25a²/25) - (24a²/25)*cos(theta1)a² = a² - (24a²/25)*cos(theta1)Subtract a² from both sides:0 = - (24a²/25)*cos(theta1)Which implies cos(theta1) = 0, so theta1 = 90 degrees.Wait, that suggests that the angle between AO and BO is 90 degrees. Similarly, in triangle COD, we can do the same:CD² = CO² + OD² - 2*CO*OD*cos(theta2)CO = 3b/5, OD = 4b/5, CD = bb² = (9b²/25) + (16b²/25) - 2*(3b/5)*(4b/5)*cos(theta2)Similarly:b² = 25b²/25 - (24b²/25)*cos(theta2)b² = b² - (24b²/25)*cos(theta2)Which also gives cos(theta2) = 0, so theta2 = 90 degrees.Therefore, the angles between the segments of the diagonals at the intersection point are 90 degrees. Therefore, the angle theta between the diagonals is 90 degrees. Wait, but in triangle AOB, the angle theta1 is the angle between AO and BO, which are parts of the diagonals. Similarly, in triangle COD, theta2 is the angle between CO and OD. Since theta1 and theta2 are both 90 degrees, that suggests that the diagonals intersect at 90 degrees. Therefore, theta = 90 degrees.Therefore, the angle between the diagonals is 90 degrees, so the area is (1/2)*d1*d2*sin(theta) = (1/2)*6*8*sin(90°) = 24*1 = 24. Therefore, the area is 24. But wait, according to our earlier formula, the area is also 5h. Therefore, 5h = 24 ⇒ h = 24/5 = 4.8. So is that possible?But let's verify if this is correct. Let's check if the angle between the diagonals is indeed 90 degrees.Wait, when I applied the law of cosines to triangle AOB, I found that theta1 = 90 degrees, and similarly theta2 = 90 degrees. However, in reality, the angle between the diagonals is the same throughout. Wait, actually, if the diagonals intersect at point O, then the angle between AO and BO is theta1, and the angle between CO and OD is theta2. If theta1 and theta2 are both 90 degrees, then the diagonals are perpendicular to each other. Because when two lines intersect, the vertically opposite angles are equal, so if theta1 is 90 degrees, theta2 would also be 90 degrees. Therefore, the diagonals are perpendicular.Therefore, in this trapezoid, the diagonals are perpendicular (since theta = 90 degrees), so the area is indeed (1/2)*6*8 = 24. Therefore, the area of the trapezoid is 24. But let's cross-verify this with another approach.Alternatively, using the midline and height. Since a + b = 10, and area = 5h, if the area is 24, then h = 24/5 = 4.8. Let's see if this height makes sense with the diagonals.Let’s take a coordinate system again. Let’s assume that the trapezoid is such that the lower base is a, upper base is b, with a + b = 10. Let's suppose that a and b are such that |b - a| = 10. Wait, but earlier that led to a contradiction. Wait, but if the diagonals are perpendicular, maybe there's a different relationship.Wait, but if the diagonals are perpendicular, there might be a formula for the area in terms of the midline. Wait, midline is 5, which is (a + b)/2. If the diagonals are perpendicular, then area is (1/2)*d1*d2. But (1/2)*6*8 = 24, which matches. But does that formula hold for trapezoids with perpendicular diagonals?Wait, the formula (1/2)*d1*d2 is for the area when the diagonals are perpendicular, regardless of the type of quadrilateral. So yes, if a quadrilateral (including trapezoid) has perpendicular diagonals, then its area is (1/2)*d1*d2. Therefore, in this case, since diagonals are 6 and 8 and they are perpendicular, the area is indeed 24. Therefore, the answer is 24.But wait, let's confirm once again. Suppose we have a trapezoid with perpendicular diagonals of lengths 6 and 8. Then the area is 24. The midline is 5, which is (a + b)/2. So a + b = 10, which is consistent. But what about the height? If area is 24 = (1/2)*(a + b)*h = 5h, so h = 24/5 = 4.8. Let's see if we can find such a trapezoid.Alternatively, take an example. Let’s suppose a = 0. But no, a and b can't be zero. Wait, but earlier when we tried to solve a + b = 10 and |a - b| = 10, it led to a = 10, b = 0 or vice versa, which isn't possible. However, in reality, the problem didn't state that |a - b| = 10. Wait, in the coordinate system approach, we ended up with |b - a| = 10, but that led to a contradiction, unless there's a miscalculation. Wait, maybe my mistake was in the coordinate system.Wait, let's go back. The distance between the midpoints of the legs is equal to the midline. Wait, but in my coordinate system, the midline's length was computed as |(d - a - c)/2| = 5, leading to |b - a| = 10. However, if the trapezoid is not convex, maybe the coordinates are different? Or perhaps my coordinate assignments are incorrect.Alternatively, maybe the midline is indeed 5, so a + b = 10, but the difference |a - b| is not 10. Wait, but according to the coordinate system calculation, |b - a| = 10. But how?Wait, perhaps my coordinate system is not general enough. Let me try a different approach.Let’s consider the trapezoid with bases of length a and b, and legs of length c and d. Let’s denote the height as h. The midline m = (a + b)/2 = 5, so a + b = 10. The diagonals are p = 6 and q = 8.There is a formula for the diagonals in a trapezoid. In general, the lengths of the diagonals can be found using the formula:p² + q² = c² + d² + 2abBut I need to check if this formula is correct. Wait, no, that formula might be for a parallelogram. In a parallelogram, the sum of the squares of the diagonals equals twice the sum of the squares of the sides: p² + q² = 2(a² + b²). But in a trapezoid, the formula is different.Alternatively, I found a formula that in a trapezoid, the sum of the squares of the diagonals equals the sum of the squares of all four sides plus 4h². Wait, no, perhaps that's not accurate.Alternatively, another approach. Let's denote the legs as the non-parallel sides. Let’s project the upper base onto the lower base. The difference in the lengths of the bases is |a - b|. The projection of each leg onto the base would contribute to this difference. However, this might complicate things.Alternatively, use the formula for the diagonals in terms of the bases, legs, and height. For a trapezoid with bases a and b, legs c and d, and height h, the diagonals p and q can be expressed using the following formulas:p² = h² + (a + x)²q² = h² + (a - x)²Where x is the horizontal distance between the foot of the height from the upper base to the lower base. Wait, not sure.Alternatively, consider the trapezoid as two right triangles and a rectangle. If we extend the legs to form a right triangle, but since the legs can be of unequal length, this might not hold.Alternatively, use vectors. Let’s model the trapezoid with vectors. Let’s set point A at the origin (0, 0), base AB along the x-axis with length a, so point B is at (a, 0). The upper base CD is parallel to AB, with length b, and height h. Let’s denote vector AD as a leg, which has components (d, h), where d is the horizontal component. Similarly, vector BC is another leg, which would have components (e, h), where e is the horizontal component. The difference between the bases is a - b = d + e. Since the upper base CD is of length b, the coordinates of point C would be (d, h), and point D would be (d + b, h). Wait, maybe this is a better coordinate system.Let me formalize this.Let’s place the trapezoid with lower base AB of length a at (0, 0) to (a, 0). The upper base CD of length b is at height h, so points C and D are at (d, h) and (d + b, h) respectively. The legs are AD and BC. Vector AD goes from A(0,0) to D(d + b, h), so the components are (d + b, h). Vector BC goes from B(a, 0) to C(d, h), so the components are (d - a, h). The legs AD and BC have lengths sqrt[(d + b)^2 + h²] and sqrt[(d - a)^2 + h²] respectively.But since the problem doesn't give information about the legs, perhaps this isn't helpful. However, the diagonals are AC and BD. Let's compute their lengths.Diagonal AC connects A(0, 0) to C(d, h). Its length is sqrt[d² + h²] = 6.Diagonal BD connects B(a, 0) to D(d + b, h). Its length is sqrt[(d + b - a)^2 + h²] = 8.Additionally, the midline is 5, so (a + b)/2 = 5 ⇒ a + b = 10.Also, since the upper base CD has length b, the horizontal distance from C to D is b, so the coordinates are set as such.Now, we have three equations:1. sqrt[d² + h²] = 6 ⇒ d² + h² = 36.2. sqrt[(d + b - a)^2 + h²] = 8 ⇒ (d + b - a)^2 + h² = 64.3. a + b = 10.We need a fourth equation. Let's see. The legs are AD and BC.But vector AD is from A(0,0) to D(d + b, h), so length AD = sqrt[(d + b)^2 + h²].Vector BC is from B(a,0) to C(d, h), so length BC = sqrt[(d - a)^2 + h²].But since we don't have information about the legs, maybe we can relate d and the difference between the bases. The horizontal projection difference between the two bases is a - b = (d + b) - d = b? Wait, no. Wait, the lower base is from 0 to a, the upper base is from d to d + b. The horizontal overlap between the bases is from max(0, d) to min(a, d + b). The difference in the projections is determined by the horizontal shift d. The horizontal component that is not overlapping is |a - (d + b - d)| = |a - b|. Wait, maybe this is getting too complicated.Alternatively, note that the horizontal distance between the two bases is |d - (a - b)/2| or something like that. Wait, no.Alternatively, consider that the difference in the lengths of the bases is |a - b|, and this difference is related to the horizontal components of the legs. If the trapezoid was a rectangle, the legs would be vertical, so d = 0, and a = b. But since it's a trapezoid, the legs have horizontal components. The sum of the horizontal components of the legs equals |a - b|. But since there are two legs, each contributing a horizontal component. Let’s denote the horizontal component of AD as x, and the horizontal component of BC as y. Then x + y = |a - b|. However, depending on the direction of the legs, x and y could be positive or negative. But in the coordinate system I set up earlier, the horizontal component of AD is d + b, and the horizontal component of BC is d - a. Wait, no, that's the vector components, but in reality, the horizontal difference between the bases is such that the upper base is shifted by some amount relative to the lower base. Let me think.Alternatively, the horizontal shift between the two bases is d. So if the lower base starts at 0, the upper base starts at d. Therefore, the horizontal shift is d. The difference in the lengths of the bases is |a - b|. Therefore, if the upper base is shifted to the right by d, then the legs have horizontal components of d and (a - b - d). Wait, maybe not.Alternatively, from the coordinates, the lower base is from 0 to a, and the upper base is from d to d + b. The horizontal shift between the two bases is d. If d is positive, the upper base is shifted to the right; if negative, to the left.The total horizontal distance between the ends of the bases is |a - (d + b - d)| = |a - b|. Wait, that doesn't make sense. Wait, the right end of the lower base is at a, and the right end of the upper base is at d + b. The horizontal distance between these two points is |d + b - a|. Similarly, the left end of the lower base is at 0, and the left end of the upper base is at d. The horizontal distance between them is |d - 0| = |d|. Therefore, the total horizontal shift is determined by d and |d + b - a|.But I think this approach is getting too convoluted. Let me get back to the equations I had earlier:1. d² + h² = 36.2. (d + b - a)^2 + h² = 64.3. a + b = 10.Let me try to substitute a = 10 - b into the second equation.So, equation 2 becomes: (d + b - (10 - b))^2 + h² = 64 ⇒ (d + 2b - 10)^2 + h² = 64.But from equation 1, h² = 36 - d². Substitute into equation 2:(d + 2b - 10)^2 + (36 - d²) = 64.Expand (d + 2b - 10)^2:= d² + 4b² + 100 + 4bd - 20d - 40b.Wait, no, wait. (d + 2b - 10)^2 = [d + (2b - 10)]^2 = d² + 2*d*(2b - 10) + (2b - 10)^2.Which is:= d² + 4bd - 20d + 4b² - 40b + 100.Therefore, equation 2 becomes:d² + 4bd - 20d + 4b² - 40b + 100 + 36 - d² = 64.Simplify:The d² terms cancel: d² - d² = 0.Remaining terms:4bd - 20d + 4b² - 40b + 100 + 36 = 64.Combine constants: 100 + 36 = 136.So:4bd - 20d + 4b² - 40b + 136 = 64.Subtract 64:4bd - 20d + 4b² - 40b + 72 = 0.Divide entire equation by 4:bd - 5d + b² - 10b + 18 = 0.So, we have:b² + bd - 10b -5d + 18 = 0.Hmm, now we have an equation with variables b and d, and we know that a + b = 10 ⇒ a = 10 - b. But how to proceed from here?Perhaps we can express d in terms of b from equation 1. From equation 1: d² = 36 - h². Wait, no, equation 1 is d² + h² = 36, so h² = 36 - d². Not helpful.Alternatively, we can use another equation. Wait, the problem gives diagonals 6 and 8, midline 5. The earlier approach where the diagonals are perpendicular gives area 24, which seems to check out, but when we tried coordinate system, we encountered inconsistencies. However, the angle between the diagonals came out as 90 degrees through the law of cosines approach, leading to area 24, which seems plausible.Alternatively, maybe the answer is indeed 24, and the contradictions in the coordinate system arise from the fact that such a trapezoid cannot exist unless the angle between the diagonals is 90 degrees. Wait, but if the calculations lead to a contradiction in the coordinate system, that suggests that the trapezoid with these properties can only exist if the diagonals are perpendicular. Therefore, the area is 24.Alternatively, let's check with actual numbers. Suppose the midline is 5, so a + b = 10. Let's take a = 10 - b. If diagonals are 6 and 8, and they are perpendicular, area is 24. Let’s see if such a trapezoid can exist.Assume diagonals are perpendicular. Then h = 24/5 = 4.8. Let’s check if with h = 4.8, a + b = 10, and diagonals 6 and 8, whether the coordinates can be set up.Take a coordinate system with A(0,0), B(a,0), D(d, h), C(c, h). Then AC = 6: sqrt(c² + h²) = 6 ⇒ c² + h² = 36. BD = 8: sqrt((d - a)^2 + h²) = 8 ⇒ (d - a)^2 + h² = 64.Also, since CD has length b, c = d - b.Also, midline is 5 ⇒ (a + b)/2 = 5 ⇒ a + b = 10.So, let's assume a = 10 - b.Then, c = d - b.From AC: sqrt(c² + h²) = 6 ⇒ (d - b)^2 + h² = 36.From BD: sqrt((d - a)^2 + h²) = 8 ⇒ (d - (10 - b))^2 + h² = 64 ⇒ (d - 10 + b)^2 + h² = 64.So we have two equations:1. (d - b)^2 + h² = 36.2. (d + b - 10)^2 + h² = 64.Subtract equation 1 from equation 2:[(d + b - 10)^2 - (d - b)^2] = 64 - 36 = 28.Expand both squares:= [ (d + b - 10)^2 - (d - b)^2 ]= [ (d^2 + 2d(b - 10) + (b - 10)^2 ) - (d^2 - 2db + b^2 ) ]Wait, alternatively, using the difference of squares:= [ (d + b - 10 - (d - b)) * (d + b - 10 + (d - b)) ]= [ (d + b -10 - d + b) * (d + b -10 + d - b) ]= [ (2b - 10) * (2d - 10) ] = 28.So:(2b - 10)(2d - 10) = 28.Divide both sides by 4:(b - 5)(d - 5) = 7.So, (b - 5)(d - 5) = 7.But we also have from equation 1:(d - b)^2 + h² = 36.And h = 24/5 = 4.8, so h² = (24/5)^2 = 576/25 = 23.04.Then, (d - b)^2 = 36 - 23.04 = 12.96.So, (d - b)^2 = 12.96 ⇒ d - b = ±3.6.Therefore, d = b ± 3.6.So, now we have two possibilities:1. d = b + 3.6.2. d = b - 3.6.Let’s substitute into (b - 5)(d - 5) = 7.Case 1: d = b + 3.6.Then, (b - 5)(b + 3.6 - 5) = 7 ⇒ (b - 5)(b - 1.4) = 7.Let’s expand this:b² - 1.4b -5b + 7 = 7 ⇒ b² - 6.4b + 7 = 7 ⇒ b² - 6.4b = 0 ⇒ b(b - 6.4) = 0.So, b = 0 or b = 6.4.But since a + b = 10 and a must be positive, b can’t be 0. So, b = 6.4. Then, a = 10 - 6.4 = 3.6.Then, d = 6.4 + 3.6 = 10.0.Check if this works.Then, check equation 1:(d - b)^2 + h² = (10.0 - 6.4)^2 + 23.04 = (3.6)^2 + 23.04 = 12.96 + 23.04 = 36. Correct.Equation 2:(d + b - 10)^2 + h² = (10.0 + 6.4 - 10)^2 + 23.04 = (6.4)^2 + 23.04 = 40.96 + 23.04 = 64. Correct.Therefore, this works. So, a = 3.6, b = 6.4, d = 10.0.Check the coordinates:A(0,0), B(3.6, 0), D(10.0, 4.8), C(10.0 - 6.4, 4.8) = (3.6, 4.8).Wait, then CD is from (3.6, 4.8) to (10.0, 4.8), which is length 10.0 - 3.6 = 6.4 = b. Correct.AB is from (0,0) to (3.6,0), length 3.6 = a. Correct.Diagonals:AC: from (0,0) to (3.6,4.8). Length sqrt(3.6² + 4.8²) = sqrt(12.96 + 23.04) = sqrt(36) = 6. Correct.BD: from (3.6,0) to (10.0,4.8). Length sqrt((10.0 - 3.6)^2 + (4.8 - 0)^2) = sqrt(6.4² + 4.8²) = sqrt(40.96 + 23.04) = sqrt(64) = 8. Correct.Midline: (a + b)/2 = (3.6 + 6.4)/2 = 10/2 = 5. Correct.Therefore, this is a valid trapezoid with the given properties. The height is 4.8, so area is 5*4.8 = 24. Therefore, the area is indeed 24.Case 2: d = b - 3.6.Then, (b - 5)(b - 3.6 - 5) = 7 ⇒ (b -5)(b - 8.6) = 7.Expanding:b² - 8.6b -5b + 43 = 7 ⇒ b² -13.6b + 43 = 7 ⇒ b² -13.6b + 36 = 0.Solve quadratic equation:b = [13.6 ± sqrt(13.6² - 4*1*36)] / 2.Compute discriminant:13.6² = 184.96, 4*1*36 = 144.Discriminant = 184.96 - 144 = 40.96.sqrt(40.96) = 6.4.Thus, b = [13.6 ± 6.4]/2.So, two solutions:b = (13.6 + 6.4)/2 = 20/2 = 10.Or b = (13.6 - 6.4)/2 = 7.2/2 = 3.6.If b = 10, then a = 10 - 10 = 0. Invalid.If b = 3.6, then a = 10 - 3.6 = 6.4. Then d = 3.6 - 3.6 = 0. Then check equation 1:(d - b)^2 + h² = (0 - 3.6)^2 + 23.04 = 12.96 + 23.04 = 36. Correct.Equation 2:(d + b - 10)^2 + h² = (0 + 3.6 - 10)^2 + 23.04 = (-6.4)^2 + 23.04 = 40.96 + 23.04 = 64. Correct.Therefore, this also works. So, a = 6.4, b = 3.6, d = 0.Coordinates:A(0,0), B(6.4,0), D(0,4.8), C(0 - 3.6,4.8) = (-3.6,4.8).Wait, CD is from (-3.6,4.8) to (0,4.8), length 0 - (-3.6) = 3.6 = b. Correct.AB is from (0,0) to (6.4,0), length 6.4 = a. Correct.Diagonals:AC: from (0,0) to (-3.6,4.8). Length sqrt((-3.6)^2 + 4.8^2) = sqrt(12.96 + 23.04) = sqrt(36) = 6. Correct.BD: from (6.4,0) to (0,4.8). Length sqrt((0 - 6.4)^2 + (4.8 - 0)^2) = sqrt(40.96 + 23.04) = sqrt(64) = 8. Correct.Midline: (6.4 + 3.6)/2 = 10/2 = 5. Correct.Therefore, this is another valid trapezoid with the given properties, mirrored compared to the first one. The area is still 5*4.8 = 24.Therefore, in both cases, the area is 24, confirming that the area of the trapezoid is indeed 24.Final AnswerThe area of the trapezoid is boxed{24}.</think>Given a trapezoid with diagonals of lengths 6 and 8, and a midline of 5, we need to find its area.1. Midline Calculation: The midline of a trapezoid is the average of the lengths of the two bases. Given the midline is 5, we have: [ frac{a + b}{2} = 5 implies a + b = 10 ]2. Area Formula: The area of a trapezoid is given by: [ text{Area} = frac{1}{2} times (a + b) times h ] Substituting (a + b = 10), the area simplifies to: [ text{Area} = 5h ]3. Diagonals and Angle: The area of a quadrilateral with diagonals (d_1) and (d_2) intersecting at an angle (theta) is: [ text{Area} = frac{1}{2} times d_1 times d_2 times sin(theta) ] For the given trapezoid, (d_1 = 6), (d_2 = 8), and we need to determine (theta).4. Law of Cosines and Perpendicular Diagonals: Using the law of cosines on the triangles formed by the diagonals, we found that the angle (theta) between the diagonals is 90 degrees. This is because the calculations showed that the angle between the segments of the diagonals at their intersection point is 90 degrees.5. Area Calculation: With (theta = 90^circ), the area is: [ text{Area} = frac{1}{2} times 6 times 8 times sin(90^circ) = frac{1}{2} times 6 times 8 times 1 = 24 ]6. Verification: The height (h) of the trapezoid can be found using the area formula (5h = 24), giving (h = frac{24}{5} = 4.8). This height was verified to be consistent with the given diagonals and midline through coordinate geometry.Thus, the area of the trapezoid is (boxed{24}).]

🔑:Let h be the height of trapezoid ABCD with bases AD and BC and diagonals AC=6 and BD=8, and let l be the midline of the trapezoid. Draw a line through vertex C parallel to diagonal BD until it intersects the extension of base AD at point M. Then quadrilateral BCMD is a parallelogram, soCM = BD = 8, DM = BC, AM = AD + DM = AD + BC = 2l = 10.Thus, triangle ACM is a right triangle ( AM^2 = AC^2 + CM^2 ). Its area is half the product oftext{ the legs, i.e., } S_{triangle ACM} = frac{1}{2} AC cdot CM = frac{1}{2} cdot 6 cdot 8 = 24. text{ Therefore, } S_{ABCD} = l h = frac{1}{2}(AD + BC) h = frac{1}{2}(AD + DM) cdot h = = frac{1}{2}(AD + BC) cdot h = frac{1}{2} AM cdot h = S_{triangle ACM} = 24.## Answer24.00Submit a comment

🧠:[<think>Okay, so I need to simplify the expression (1 - a/(a+1)) divided by (a² - 1)/(a² + 2a + 1), where a is sqrt(2) + 1. Let me start by understanding each part of the expression step by step.First, let me look at the numerator of the main fraction: 1 - a/(a+1). I think I need to combine these two terms into a single fraction. To do that, I should find a common denominator. The denominators here are 1 and (a+1), so the common denominator would be (a+1). So, rewriting 1 as (a+1)/(a+1), the numerator becomes (a+1)/(a+1) - a/(a+1). Combining these, we subtract the numerators: (a + 1 - a)/ (a + 1). The a terms cancel out, leaving (1)/(a + 1). So, the entire numerator simplifies to 1/(a + 1). Okay, that part seems manageable.Now, the denominator of the main fraction is (a² - 1)/(a² + 2a + 1). Let me look at both the numerator and the denominator here. Starting with the denominator of the main fraction: (a² - 1). That looks like a difference of squares, which factors into (a - 1)(a + 1). Then, the denominator (a² + 2a + 1) is a perfect square trinomial. It factors into (a + 1)². So, substituting these factored forms back in, the denominator of the main fraction becomes (a - 1)(a + 1)/(a + 1)². Then, simplifying this fraction, we can cancel one (a + 1) from the numerator and denominator, leaving (a - 1)/(a + 1). So now, the entire original expression simplifies to [1/(a + 1)] divided by [(a - 1)/(a + 1)]. Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes [1/(a + 1)] * [(a + 1)/(a - 1)]. Here, the (a + 1) terms cancel out, leaving 1/(a - 1). Okay, so after simplifying the entire expression, we end up with 1 divided by (a - 1). Now, we need to substitute a = sqrt(2) + 1 into this simplified expression. Substituting, we get 1/[(sqrt(2) + 1) - 1]. The +1 and -1 cancel out in the denominator, leaving 1/sqrt(2). But usually, we rationalize the denominator when there's a square root in the denominator. To rationalize 1/sqrt(2), we multiply numerator and denominator by sqrt(2): (1 * sqrt(2)) / (sqrt(2) * sqrt(2)) = sqrt(2)/2. So, the final evaluated result should be sqrt(2)/2. Wait, let me double-check my steps to make sure I didn't make a mistake. First, simplifying the numerator:1 - a/(a + 1) becomes (a + 1 - a)/(a + 1) = 1/(a + 1). That's correct.Denominator of the main fraction:(a² - 1)/(a² + 2a + 1) factors into (a - 1)(a + 1)/(a + 1)^2, which simplifies to (a - 1)/(a + 1). That's right.So the entire expression becomes [1/(a + 1)] / [(a - 1)/(a + 1)] = [1/(a + 1)] * [ (a + 1)/(a - 1) ] = 1/(a - 1). Correct.Then substituting a = sqrt(2) + 1:a - 1 = sqrt(2) + 1 - 1 = sqrt(2). So 1/(sqrt(2)) becomes sqrt(2)/2 after rationalization. Everything checks out. So the simplified expression is 1/(a - 1), and substituting a = sqrt(2) + 1 gives sqrt(2)/2. Therefore, the answer should be sqrt(2)/2. Alternatively, if I didn't rationalize the denominator, it would be 1/sqrt(2), but since rationalizing is standard, sqrt(2)/2 is the preferred form. I don't see any mistakes in my steps, so I think that's the correct answer.Final Answerboxed{dfrac{sqrt{2}}{2}}</think>First, simplify the numerator of the main fraction:[1 - frac{a}{a+1} = frac{(a+1) - a}{a+1} = frac{1}{a+1}]Next, simplify the denominator of the main fraction:[frac{a^2 - 1}{a^2 + 2a + 1} = frac{(a-1)(a+1)}{(a+1)^2} = frac{a-1}{a+1}]Now, divide the simplified numerator by the simplified denominator:[frac{frac{1}{a+1}}{frac{a-1}{a+1}} = frac{1}{a+1} times frac{a+1}{a-1} = frac{1}{a-1}]Substitute ( a = sqrt{2} + 1 ):[frac{1}{(sqrt{2} + 1) - 1} = frac{1}{sqrt{2}}]Rationalize the denominator:[frac{1}{sqrt{2}} times frac{sqrt{2}}{sqrt{2}} = frac{sqrt{2}}{2}]Thus, the final answer is:[boxed{dfrac{sqrt{2}}{2}}],<think>Okay, so I need to simplify this expression first: (1 - a/(a+1)) divided by ((a² - 1)/(a² + 2a + 1)), and then evaluate it when a is sqrt(2) + 1. Alright, let's take it step by step. First, let's look at the numerator of the big fraction: 1 - a/(a+1). Maybe I can combine those terms by getting a common denominator. The 1 can be written as (a+1)/(a+1), right? So then it would be (a+1)/(a+1) - a/(a+1). That simplifies to [(a+1) - a]/(a+1). Let's check that: (a + 1 - a)/ (a + 1) = 1/(a +1). So the numerator simplifies to 1/(a +1). That seems right. Let me double-check. If I have 1 minus a/(a+1), multiplying numerator and denominator, yeah, that works. So numerator is 1/(a+1).Now the denominator of the entire expression is ((a² -1)/(a² + 2a +1)). Let me look at that. The numerator here is a² -1, which is a difference of squares, so it factors into (a -1)(a +1). The denominator here is a² + 2a +1, which is a perfect square, so it factors into (a +1)^2. So that simplifies the denominator part to (a -1)(a +1)/(a +1)^2. Then the (a +1) terms cancel out, leaving (a -1)/(a +1). So the entire denominator of the main expression simplifies to (a -1)/(a +1). So now the entire original expression is (1/(a +1)) divided by ((a -1)/(a +1)). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes (1/(a +1)) * ((a +1)/(a -1)). The (a +1) terms cancel out, leaving 1/(a -1). So after simplifying the entire expression, it's just 1 divided by (a -1). Now, we need to evaluate this when a = sqrt(2) + 1. So substitute a into the simplified expression:1/( (sqrt(2) + 1) - 1 ) = 1/(sqrt(2) + 1 - 1) = 1/sqrt(2). Wait, that seems straightforward. So 1 divided by sqrt(2). But usually, we rationalize the denominator, right? So multiplying numerator and denominator by sqrt(2) gives sqrt(2)/2. So the final answer would be sqrt(2)/2. Let me double-check my steps to make sure I didn't make a mistake. Starting with the numerator: 1 - a/(a +1). Converted 1 to (a +1)/(a +1), subtracted a/(a +1), got 1/(a +1). That seems correct. Then the denominator: (a² -1)/(a² + 2a +1). Factored numerator as (a -1)(a +1) and denominator as (a +1)^2. Then simplified to (a -1)/(a +1). Then divided the two, which cancels (a +1) terms, giving 1/(a -1). Substitute a = sqrt(2) +1. Then (sqrt(2) +1 -1) is sqrt(2). So 1/sqrt(2) becomes sqrt(2)/2. Wait, but let me check if there's an alternative way that might reveal a different result. Maybe I made a mistake in simplifying? Let's try plugging in a = sqrt(2) +1 directly into the original expression without simplifying first, just to see if we get the same answer. That might be a good check.Original expression: (1 - a/(a +1)) / ((a² -1)/(a² + 2a +1))First, compute 1 - a/(a +1) when a = sqrt(2) +1.Compute a +1: (sqrt(2) +1) +1 = sqrt(2) +2. So a/(a +1) is (sqrt(2)+1)/(sqrt(2) +2). Then 1 - that is 1 - (sqrt(2)+1)/(sqrt(2)+2). Let's compute this:Convert 1 to (sqrt(2) +2)/(sqrt(2) +2). Then subtract (sqrt(2) +1)/(sqrt(2) +2). So [(sqrt(2)+2) - (sqrt(2) +1)]/(sqrt(2) +2) = (sqrt(2) +2 - sqrt(2) -1)/ (sqrt(2)+2) = (1)/ (sqrt(2) +2). So the numerator is 1/(sqrt(2)+2).Now the denominator of the original expression is (a² -1)/(a² +2a +1). Let's compute that when a = sqrt(2) +1.First, compute a²: (sqrt(2) +1)^2 = (2 + 2sqrt(2) +1) = 3 + 2sqrt(2). So a² -1 = (3 + 2sqrt(2)) -1 = 2 + 2sqrt(2). a² + 2a +1: Let's compute this. a² is already 3 + 2sqrt(2). Then 2a is 2*(sqrt(2)+1) = 2sqrt(2) +2. Adding 1 gives (3 + 2sqrt(2)) + (2sqrt(2) +2) +1? Wait, no. Wait, a² +2a +1 is (a +1)^2. Since a = sqrt(2) +1, then a +1 = sqrt(2) +2, so (a +1)^2 = (sqrt(2) +2)^2 = 2 +4sqrt(2) +4 = 6 +4sqrt(2). Alternatively, computing directly:a² +2a +1 = (3 + 2sqrt(2)) + 2*(sqrt(2) +1) +1 = 3 + 2sqrt(2) + 2sqrt(2) +2 +1 = (3+2+1) + (2sqrt(2)+2sqrt(2)) = 6 +4sqrt(2). So denominator is (a² -1)/(a² +2a +1) = (2 + 2sqrt(2))/(6 +4sqrt(2)).So the entire original expression is [1/(sqrt(2)+2)] divided by [(2 + 2sqrt(2))/(6 +4sqrt(2))]. Dividing by a fraction is multiplying by its reciprocal, so:[1/(sqrt(2)+2)] * [(6 +4sqrt(2))/(2 + 2sqrt(2))]Simplify numerator and denominator:First, factor numerator and denominator if possible. Let's look at 6 +4sqrt(2) and 2 + 2sqrt(2). Factor 2 from denominator: 2*(1 + sqrt(2)). Factor 2 from numerator: 6 +4sqrt(2) = 2*(3 + 2sqrt(2)). So now we have:[1/(sqrt(2)+2)] * [2*(3 + 2sqrt(2)) / (2*(1 + sqrt(2)))] The 2's cancel out, so:[1/(sqrt(2)+2)] * [(3 + 2sqrt(2))/(1 + sqrt(2))]Now, let's compute (3 + 2sqrt(2))/(1 + sqrt(2)). Maybe rationalize the denominator here. Multiply numerator and denominator by (1 - sqrt(2)):Numerator: (3 + 2sqrt(2))(1 - sqrt(2)) = 3*1 +3*(-sqrt(2)) +2sqrt(2)*1 +2sqrt(2)*(-sqrt(2)) = 3 -3sqrt(2) +2sqrt(2) -2*(2) = 3 -sqrt(2) -4 = (-1 -sqrt(2))Denominator: (1 + sqrt(2))(1 - sqrt(2)) =1 - (sqrt(2))^2 =1 -2 = -1.So (3 + 2sqrt(2))/(1 + sqrt(2)) = (-1 -sqrt(2))/(-1) = (1 + sqrt(2))/1 =1 + sqrt(2)So now we have:[1/(sqrt(2)+2)] * (1 + sqrt(2)) Note that 1 + sqrt(2) is the same as sqrt(2) +1. So let's write it as [1/(sqrt(2)+2)] * (sqrt(2) +1)But sqrt(2) +1 and sqrt(2)+2: Wait, sqrt(2)+2 is (sqrt(2) +1) +1. Hmm, maybe factor something here. Alternatively, multiply out:[ (sqrt(2) +1) ] / (sqrt(2) +2 )Multiply numerator and denominator by (sqrt(2) -2) to rationalize? Wait, usually we multiply by conjugate. The denominator is sqrt(2) +2, so the conjugate would be sqrt(2) -2. Let's try that:Numerator: (sqrt(2) +1)(sqrt(2) -2) = sqrt(2)*sqrt(2) + sqrt(2)*(-2) +1*sqrt(2) +1*(-2) = 2 -2sqrt(2) + sqrt(2) -2 = (2 -2) + (-2sqrt(2)+sqrt(2)) = 0 -sqrt(2) = -sqrt(2)Denominator: (sqrt(2) +2)(sqrt(2) -2) = (sqrt(2))^2 - (2)^2 = 2 -4 = -2So the result is (-sqrt(2))/(-2) = sqrt(2)/2So the entire expression evaluates to sqrt(2)/2. Which matches the simplified version where we had 1/(a -1) =1/sqrt(2) = sqrt(2)/2. So both methods give the same result, so that's good. Therefore, the final answer is sqrt(2)/2. But wait, let me check once more. Because when I substituted directly, I had to go through a lot of steps, but both ways gave me sqrt(2)/2. So seems consistent. So I think that's the correct answer. So the key was simplifying the original expression first to 1/(a -1), then substituting a = sqrt(2)+1, leading to 1/sqrt(2), rationalizing gives sqrt(2)/2. Alternatively, substituting directly, after some work, also gives sqrt(2)/2. So the answer is confirmed. Final Answerboxed{dfrac{sqrt{2}}{2}}</think>First, simplify the given expression ((1 - frac{a}{a+1}) div frac{a^2 - 1}{a^2 + 2a + 1}):1. Simplify the numerator (1 - frac{a}{a+1}): [ 1 - frac{a}{a+1} = frac{(a+1) - a}{a+1} = frac{1}{a+1} ]2. Simplify the denominator (frac{a^2 - 1}{a^2 + 2a + 1}): [ frac{a^2 - 1}{a^2 + 2a + 1} = frac{(a-1)(a+1)}{(a+1)^2} = frac{a-1}{a+1} ]3. Divide the simplified numerator by the simplified denominator: [ frac{frac{1}{a+1}}{frac{a-1}{a+1}} = frac{1}{a+1} times frac{a+1}{a-1} = frac{1}{a-1} ]Next, substitute (a = sqrt{2} + 1) into the simplified expression (frac{1}{a-1}):1. Calculate (a - 1): [ (sqrt{2} + 1) - 1 = sqrt{2} ]2. Substitute into the expression: [ frac{1}{sqrt{2}} ]3. Rationalize the denominator: [ frac{1}{sqrt{2}} times frac{sqrt{2}}{sqrt{2}} = frac{sqrt{2}}{2} ]Thus, the final answer is (boxed{dfrac{sqrt{2}}{2}}).]

🔑:To simplify and then evaluate the given expression (1-frac{a}{a+1})div frac{{a}^{2}-1}{{a}^{2}+2a+1} with a=sqrt{2}+1, we proceed as follows:First, we simplify the expression in the numerator and the denominator separately:1. Simplify the numerator: (1-frac{a}{a+1})- We can write 1 as frac{a+1}{a+1} to have a common denominator: 1-frac{a}{a+1} = frac{a+1}{a+1} - frac{a}{a+1}- Simplifying this gives: frac{a+1-a}{a+1} = frac{1}{a+1}2. Simplify the denominator: frac{{a}^{2}-1}{{a}^{2}+2a+1}- Notice that {a}^{2}-1 is a difference of squares, which can be factored as (a+1)(a-1).- The denominator {a}^{2}+2a+1 is a perfect square, which can be written as (a+1)^{2}.- Thus, the denominator simplifies to: frac{(a+1)(a-1)}{(a+1)^{2}}Putting the simplified numerator and denominator together, we have:(frac{1}{a+1})div frac{(a+1)(a-1)}{(a+1)^{2}}3. Simplify the division:- Dividing by a fraction is the same as multiplying by its reciprocal: frac{1}{a+1} cdot frac{(a+1)^{2}}{(a+1)(a-1)}- Simplifying this gives: frac{1}{a-1}Now, substituting a=sqrt{2}+1 into the simplified expression:frac{1}{sqrt{2}+1-1} = frac{1}{sqrt{2}}To make the expression rational, we can multiply the numerator and the denominator by sqrt{2}, which gives:frac{1cdotsqrt{2}}{sqrt{2}cdotsqrt{2}} = frac{sqrt{2}}{2}Therefore, the simplified and evaluated expression is boxed{frac{sqrt{2}}{2}}.

🧠:[<think>Alright, let's try to solve this problem. So, we have two Russian words: КРУГ and КУБ, both of which are cubes of some numbers. Each letter stands for a unique digit, and different letters represent different digits. Our task is to find all possible values of these numbers and explain why there are no others. First, let me understand the structure of the words. КРУГ is a four-letter word, and КУБ is a three-letter word. In terms of numbers, КРУГ is a four-digit number, and КУБ is a three-digit number. Each letter corresponds to a digit, so К, Р, У, Г, Б are all digits, with К, Р, У, Г being different from each other, and in КУБ, К, У, Б must also be different. However, since К and У are common to both words, their digits must be the same in both numbers. So, for example, if К is 5 in КРУГ, then К must also be 5 in КУБ. Similarly, У must be the same digit in both words. The other letters, Р, Г, and Б, must be different from each other and from К and У.So, the problem is to find all possible cubes where:1. КРУГ is a four-digit cube with distinct digits К, Р, У, Г.2. КУБ is a three-digit cube with distinct digits К, У, Б.3. К and У are the same digits in both numbers.4. All digits in КРУГ and КУБ are different except for К and У. So, Б must be different from Р, Г, and also different from К and У.First, let's list all three-digit cubes and four-digit cubes. Then we can check which ones share the first two digits (К and У) and satisfy the uniqueness conditions.Starting with three-digit cubes. The cube numbers between 100 and 999 inclusive. Let's find the cube roots. The cube root of 100 is approximately 4.64, and the cube root of 999 is approximately 9.99. So, the integers from 5 to 9 inclusive when cubed will give three-digit numbers. Let's list them:5³ = 1256³ = 2167³ = 3438³ = 5129³ = 729So, the three-digit cubes are 125, 216, 343, 512, 729. Let's note these down as possible candidates for КУБ.Now, moving on to four-digit cubes. The cube numbers between 1000 and 9999. The cube root of 1000 is 10, and the cube root of 9999 is approximately 21.54. So, integers from 10 to 21 inclusive when cubed will give four-digit numbers. Let's list them:10³ = 100011³ = 133112³ = 172813³ = 219714³ = 274415³ = 337516³ = 409617³ = 491318³ = 583219³ = 685920³ = 800021³ = 9261So, these are the four-digit cubes. Now, we need to see if any of these four-digit cubes can be paired with a three-digit cube such that:- The first digit of both numbers (К) is the same.- The second digit of the four-digit number (Р) is different from К and У.- The third digit of the four-digit number (У) is the same as the second digit of the three-digit cube (У).- The fourth digit of the four-digit number (Г) is different from К, Р, У.- The third digit of the three-digit cube (Б) is different from К, У, and also different from Р and Г.Let me try to approach this systematically.First, let's list all three-digit cubes and note their digits:1. 125: К=1, У=2, Б=52. 216: К=2, У=1, Б=63. 343: К=3, У=4, Б=3 (But here, Б=3 is same as К=3, so invalid since digits must be different)4. 512: К=5, У=1, Б=25. 729: К=7, У=2, Б=9So, from the three-digit cubes, the valid ones are 125, 216, 512, 729. (343 is invalid because Б=3 and К=3 are the same.)Now, let's look at the four-digit cubes and see if any of them start with the same К as the three-digit cubes and have У as the third digit (since in КРУГ, У is the third digit, and in КУБ, У is the second digit). Wait, hold on. In КУБ, which is a three-digit number, the structure is:К (hundreds place), У (tens place), Б (units place). So, У is the second digit in КУБ.In КРУГ, which is a four-digit number, the structure is:К (thousands place), Р (hundreds place), У (tens place), Г (units place). So, У is the third digit in КРУГ.Therefore, in both numbers, the letters К and У must correspond to the same digits. In КУБ, the digits are К (hundreds), У (tens), Б (units). In КРУГ, the digits are К (thousands), Р (hundreds), У (tens), Г (units).So, the mapping is:For КУБ: first digit is К, second digit is У, third digit is Б.For КРУГ: first digit is К, third digit is У, with Р and Г being the second and fourth digits respectively.Therefore, the four-digit number must have К as the first digit, У as the third digit. The three-digit number must have К as the first digit, У as the second digit. So, we need to find pairs where:- The four-digit cube starts with К and has У as the third digit.- The three-digit cube starts with К and has У as the second digit.Additionally, all digits in each number must be unique, and digits between the two numbers must not overlap except for К and У.So, let's take each three-digit cube and see if there's a four-digit cube that starts with the same К and has У as the third digit.Starting with the three-digit cubes:1. 125: К=1, У=2, Б=5We need a four-digit cube starting with 1 (К=1) and with У=2 as the third digit (i.e., the tens place). Let's check the four-digit cubes starting with 1:1000: digits are 1,0,0,0 – repeats, invalid.1331: digits are 1,3,3,1 – repeats, invalid.1728: digits are 1,7,2,8. Here, third digit is 2 (which is У=2). So, this is a candidate. Let's check:К=1, Р=7, У=2, Г=8. All digits unique. Then, in the three-digit cube 125, digits are К=1, У=2, Б=5. So, Б=5 is different from К=1, У=2, and also different from Р=7 and Г=8. So, this seems valid.So, pair 1728 (КРУГ) and 125 (КУБ). Let's note this as a possible solution.2. 216: К=2, У=1, Б=6Looking for four-digit cubes starting with 2 (К=2) and with У=1 as the third digit (tens place). Let's check four-digit cubes starting with 2:2197: digits 2,1,9,7. Third digit is 9 ≠ 1.2744: digits 2,7,4,4 – repeated 4s.So, no four-digit cube starting with 2 has third digit 1. Hence, no pair here.3. 512: К=5, У=1, Б=2Looking for four-digit cubes starting with 5 (К=5) and with У=1 as the third digit. Let's check:Four-digit cubes starting with 5:512 is three-digit, so next is 13³=2197 (starts with 2), 14³=2744 (starts with 2), 15³=3375 (starts with 3), 16³=4096 (starts with 4), 17³=4913 (starts with 4), 18³=5832 (starts with 5). Wait, 18³=5832. Let's check 5832: digits 5,8,3,2. Third digit is 3 ≠1. Next, 19³=6859 (starts with 6), 20³=8000 (starts with 8), 21³=9261 (starts with 9). So, the only four-digit cube starting with 5 is 5832, which has third digit 3. So, no four-digit cube starting with 5 has third digit 1. Therefore, no pair here.4. 729: К=7, У=2, Б=9Looking for four-digit cubes starting with 7 (К=7) and with У=2 as the third digit. Let's check four-digit cubes starting with 7:6859: starts with 6, so not relevant. Wait, 18³=5832, 19³=6859, 20³=8000, 21³=9261. Wait, none of the four-digit cubes start with 7 except 19³=6859, which starts with 6. Wait, perhaps I made a mistake here. Let me check again.10³=100011³=133112³=172813³=219714³=274415³=337516³=409617³=491318³=583219³=685920³=800021³=9261So, between 10³ and 21³, the cubes starting with 7 would be numbers between 7000 and 7999. The cube root of 7000 is approximately 19.1, and cube root of 7999 is approximately 19.99. So, only 19³ is in this range. 19³=6859, which starts with 6. Wait, that can't be. 19*19*19=6859, which is 6,859. So, actually, there are no four-digit cubes starting with 7. Because 19³ is 6859 (starts with 6), 20³ is 8000 (starts with 8). Therefore, there is no four-digit cube starting with 7. Hence, no possible pair here.So, from the three-digit cubes, only 125 (КУБ=125) can pair with a four-digit cube, which is 1728 (КРУГ=1728). Let's verify this pair.Check digits:КРУГ=1728: digits are 1,7,2,8. All unique.КУБ=125: digits are 1,2,5. All unique.Common digits: К=1, У=2. The other digits in КУБ are Б=5, which is different from К, У, and also different from Р=7 and Г=8 from КРУГ. Similarly, in КРУГ, Р=7 and Г=8 are different from each other and from К and У. So, all digits are unique across both numbers. Therefore, this is a valid pair.Now, let's check if there are any other pairs. Wait, maybe we missed some three-digit cubes? Let's re-examine the three-digit cubes. Wait, initially, we considered 5³=125, 6³=216, 7³=343, 8³=512, 9³=729. But 343 has repeating digits (3 and 4 and 3 again), so invalid. So, only 125, 216, 512, 729 are valid. And as we saw, only 125 can pair with 1728.But let's double-check to ensure we haven't missed any possibilities. Let's cross-verify.Take the four-digit cubes and check if any of them can pair with a three-digit cube.Take the four-digit cube 1728. Its digits are 1,7,2,8. So, К=1, У=2. Then, looking for a three-digit cube starting with К=1 and with У=2 as the second digit. The three-digit cubes starting with 1 are 125 (1-2-5) and 1331 (invalid due to repeating digits). So, 125 is the only one. So, that's the pair we found.Another four-digit cube: 4096 (16³). Digits 4,0,9,6. So, К=4, У=9. Now, check if there is a three-digit cube starting with 4 and with У=9 as the second digit. The three-digit cubes are 125,216,343,512,729. None start with 4. So, no pair here.Next, 5832 (18³). Digits 5,8,3,2. So, К=5, У=3. Check three-digit cubes starting with 5: 512. The second digit is 1, which doesn't match У=3. So, no pair.Next, 6859 (19³). Digits 6,8,5,9. К=6, У=5. Check three-digit cubes starting with 6: 216. The second digit is 1, not 5. So, no pair.8000 (20³). Digits 8,0,0,0. Repeats, invalid.9261 (21³). Digits 9,2,6,1. К=9, У=6. Check three-digit cubes starting with 9: 729. Second digit is 2, not 6. So, no pair.Other four-digit cubes: 1000 (invalid), 1331 (invalid), 2744 (repeats), 3375 (digits 3,3,7,5 – repeats), 4913 (digits 4,9,1,3. So, К=4, У=1. Check three-digit cubes starting with 4: none. So, invalid). 2197 (2,1,9,7. К=2, У=9. Check three-digit cubes starting with 2: 216. Second digit is 1, not 9. So, invalid). So, seems like the only valid pair is 1728 and 125.Wait a minute, let me check another four-digit cube: 4913. Digits 4,9,1,3. So, К=4, У=1. Then, is there a three-digit cube starting with 4? The three-digit cubes are up to 9³=729, so 10³=1000 is four digits. So, 7³=343, 8³=512, 9³=729. So, no three-digit cube starts with 4. Therefore, no pair here.Another four-digit cube: 2197. К=2, У=9. Check three-digit cubes starting with 2: 216. Second digit is 1, not 9. So, no.Another one: 4096. К=4, У=9. No three-digit cubes start with 4.Wait, maybe another four-digit cube: 5832. К=5, У=3. Any three-digit cubes starting with 5? 512. Second digit is 1, not 3. So, no.Wait, maybe 3375? К=3, У=7. Check three-digit cubes starting with 3: 343. But 343 has second digit 4, not 7, and also repeats digit 3. So, invalid.Similarly, 2744: digits 2,7,4,4. Repeats 4. Invalid.So, indeed, the only pair is 1728 and 125.But let me check again for КУБ=512. If К=5, У=1. Is there a four-digit cube starting with 5 with third digit 1? Let's check 5832 (third digit 3), 512 is three-digit. Are there other four-digit cubes starting with 5? 18³=5832, 19³=6859 (starts with 6). So, no. So, no four-digit cube starting with 5 with third digit 1.Similarly, for КУБ=729. К=7, У=2. Check four-digit cubes starting with 7? None, as established earlier.For КУБ=216. К=2, У=1. Check four-digit cubes starting with 2: 2197, 2744. 2197 has third digit 9, 2744 has third digit 4. Neither have third digit 1, so no pair here.So, only one possible pair: 1728 and 125.Wait, but hold on. Let's verify the digits again for 1728 and 125. КРУГ=1728: digits 1,7,2,8. All unique.КУБ=125: digits 1,2,5. All unique.Common digits: К=1, У=2. The other digits in КРУГ are 7 and 8, which are different from each other and from 1 and 2. The other digit in КУБ is 5, which is different from 1,2,7,8. So, all digits are unique across both numbers, as required.Therefore, this pair is valid.Is there another pair? Let's check another possibility. For example, maybe another three-digit cube where the second digit is shared with a four-digit cube.Wait, take КУБ=216: digits 2,1,6. So, К=2, У=1. Is there a four-digit cube starting with 2 with third digit 1? Looking at four-digit cubes starting with 2:2197: third digit 92744: third digit 4No, so no match.What about КУБ=512: digits 5,1,2. К=5, У=1. Four-digit cubes starting with 5: 5832. Third digit is 3, not 1.КУБ=729: digits 7,2,9. К=7, У=2. Four-digit cubes starting with 7: none.So, indeed, only 1728 and 125.Wait, let me check if there are any four-digit cubes with К and У that match another three-digit cube. For example, take four-digit cube 4096: digits 4,0,9,6. So, К=4, У=9. Is there a three-digit cube starting with 4? No, as we saw.Another four-digit cube: 9261. Digits 9,2,6,1. So, К=9, У=6. Check three-digit cubes starting with 9: 729. The second digit is 2, not 6. So, no.Another four-digit cube: 6859. Digits 6,8,5,9. К=6, У=5. Three-digit cubes starting with 6: 216. Second digit is 1, not 5. So, no.Another four-digit cube: 4913. Digits 4,9,1,3. К=4, У=1. Three-digit cubes starting with 4: none.So, no other pairs.Therefore, the only possible pair is КРУГ=1728 and КУБ=125.But wait, let me check another four-digit cube: 1331. Although it has repeating digits, just to confirm. К=1, У=3. But since digits repeat, it's invalid. So, no.Similarly, 1000: repeating zeros. Invalid.So, yes, only 1728 and 125.Wait, but in the four-digit cube 1728, the digits are 1,7,2,8. All unique. In the three-digit cube 125, digits are 1,2,5. All unique, and no overlaps except К and У. So, this is valid.Is there another four-digit cube that can pair with a different three-digit cube?Wait, another four-digit cube: 2197. Digits 2,1,9,7. So, К=2, У=9. Check three-digit cubes starting with 2: 216. Second digit is 1, not 9. So, no.Another four-digit cube: 5832. Digits 5,8,3,2. К=5, У=3. Three-digit cubes starting with 5: 512. Second digit is 1, not 3. No.Hmm. It seems there's no other pair. So, the only solution is 1728 and 125.Wait, but wait a second. Let me check four-digit cube 4096 again. Digits 4,0,9,6. So, К=4, У=9. Three-digit cubes starting with 4: none. But what if we consider that maybe the four-digit cube could start with a different К? Wait, no, because К is the first digit of both numbers. So, they must share the same first digit.Alternatively, is there any possibility where the same digits are used but with different letters? Wait, no, because the problem states that identical digits are replaced by the same letters, different digits by different letters. So, each letter must represent a unique digit, and different letters must represent different digits. Therefore, in both words, the letters К, У must represent the same digits, and the other letters (Р, Г, Б) must be different from each other and from К and У.Hence, after thorough checking, the only possible pair is 1728 (КРУГ) and 125 (КУБ).Therefore, the answer should be these two numbers.Final AnswerThe possible values are КРУГ = boxed{1728} and КУБ = boxed{125}.</think>To solve the problem, we need to find four-digit and three-digit cubes (КРУГ and КУБ) where each letter represents a unique digit, and different letters represent different digits. 1. Three-digit cubes (КУБ): - The valid three-digit cubes are 125, 216, 512, and 729 (343 is invalid due to repeating digits).2. Four-digit cubes (КРУГ): - The valid four-digit cubes are 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, and 9261. However, we need to check for distinct digits.3. Pairing the cubes: - We need to find pairs where the first digit (К) and the third digit (У in КРУГ) of the four-digit cube match the first digit (К) and the second digit (У in КУБ) of the three-digit cube. - After checking all possible pairs, the only valid pair is: - КРУГ = 1728 (four-digit cube, 12³) with digits 1, 7, 2, 8 (all unique). - КУБ = 125 (three-digit cube, 5³) with digits 1, 2, 5 (all unique). - Common digits: К = 1 and У = 2. The other digits (7, 8 in КРУГ and 5 in КУБ) are all unique and different from each other and from К and У.Thus, the only possible values are: Final AnswerThe possible values are КРУГ = boxed{1728} and КУБ = boxed{125}.,<think>Okay, so I need to figure out all possible values of the numbers КРУГ and КУБ where both are cubes of some numbers. Each letter represents a unique digit, and the same letter represents the same digit. The problem is in Russian, but the letters here are К, Р, У, Г, and Б. Let's translate the words: КРУГ is a 4-digit number and КУБ is a 3-digit number. Both of these numbers must be perfect cubes, and they share some digits with the constraints that different letters represent different digits. First, let me note the letters involved: К, Р, У, Г, Б. Each of these must be distinct digits. So, in both numbers КРУГ and КУБ, the letters К, У are common. Therefore, the thousands digit of КРУГ is the same as the hundreds digit of КУБ. Also, the hundreds digit of КРУГ is Р, the tens digit is У, and the units digit is Г. For КУБ, the hundreds digit is К, the tens digit is У, and the units digit is Б. Since all letters represent different digits, К, Р, У, Г, Б are all distinct.So, let me break this down step by step.First, let's consider the 3-digit number КУБ, which is a cube. Let's list all 3-digit cubes. The cubes of numbers from 5 to 9 are:5³ = 1256³ = 2167³ = 3438³ = 5129³ = 729So the possible 3-digit cubes are 125, 216, 343, 512, 729. These are the candidates for КУБ.Now, let's analyze the structure of КУБ. The hundreds digit is К, the tens digit is У, and the units digit is Б. So for each of these cubes, we can assign К, У, Б as follows:125: К=1, У=2, Б=5216: К=2, У=1, Б=6343: К=3, У=4, Б=3 → But here, Б=3 and К=3 would be the same digit, which is not allowed. So discard this.512: К=5, У=1, Б=2729: К=7, У=2, Б=9So possible candidates for КУБ are 125, 216, 512, 729. We discarded 343 because К and Б would both be 3.Now, for each of these, we need to check if we can form a 4-digit cube КРУГ with the same К and У, and digits Р and Г being different from each other and from К, У, Б.Let's take each case one by one.Case 1: КУБ = 125 (К=1, У=2, Б=5)So, К=1, У=2. Then, the 4-digit number КРУГ is 1Р2Г, where Р and Г are digits not equal to 1, 2, or 5. So possible digits for Р and Г are 0,3,4,6,7,8,9. Also, Р and Г must be distinct.We need 1Р2Г to be a perfect cube. Let's find 4-digit cubes between 1000 and 9999. The cube roots of 1000 is 10, and cube root of 9999 is approximately 21.54. So the possible cube roots are numbers from 10 to 21. However, 10³=1000, which has К=1, but let's check.Wait, but our 4-digit number starts with 1, so the cube must be between 1000 and 1999. The cube roots of 1000 is 10, and cube root of 1999 is approximately 12.6. Therefore, possible cube roots are 10, 11, 12.10³=1000: digits are 1,0,0,0. But here, Р=0, У=0, which is not possible since У is already 2. So invalid.11³=1331: digits are 1,3,3,1. Again, duplicate digits and У is supposed to be 2. Not matching.12³=1728: digits 1,7,2,8. Here, we have К=1, У=2, which matches. Then Р=7, Г=8. Check digits: К=1, Р=7, У=2, Г=8, Б=5. All digits 1,7,2,8,5 are unique. Perfect! So this is a valid solution: КРУГ=1728 (12³) and КУБ=125 (5³). So that's one possible solution.Case 2: КУБ = 216 (К=2, У=1, Б=6)Now, К=2, У=1. Then the 4-digit number КРУГ is 2Р1Г. The digits Р and Г must not be 2,1,6 and must be unique. So available digits: 0,3,4,5,7,8,9. Now, 2Р1Г is a 4-digit cube starting with 2. The cube roots of numbers from 2000 to 2999 are between approximately 12.6 and 14.4. So possible cube roots: 13, 14.13³=2197: digits 2,1,9,7. Here, К=2, У=1, which matches. Then Р=9, Г=7. Check digits: К=2, Р=9, У=1, Г=7, Б=6. All unique digits: 2,9,1,7,6. No duplicates. So this is valid. So КРУГ=2197 (13³), КУБ=216 (6³). Another solution.Case 3: КУБ=512 (К=5, У=1, Б=2)So, К=5, У=1. Then the 4-digit number is 5Р1Г. Digits Р and Г must not be 5,1,2. Available digits: 0,3,4,6,7,8,9. The cube starting with 5 would be between 5000 and 5999. The cube root of 5000 is approximately 17.1, and cube root of 5999 is approximately 18.2. Therefore, possible cube roots: 17 and 18.17³=4913: starts with 4, which is not 5. So invalid.18³=5832: digits 5,8,3,2. Here, К=5, У=3? Wait, wait. The number is 5832. But according to the structure, КРУГ=5Р1Г. But here, the tens digit is 3, but У is supposed to be 1. So this doesn't match. So 5832 is 18³, but the tens digit is 3, not 1. So invalid.Next cube: 19³=6859: starts with 6, which is not 5. So no cubes starting with 5 in this range with tens digit 1. Therefore, no solution here.Alternatively, maybe I miscalculated the cube roots. Let me check 17³: 17*17=289, 289*17=4913. Correct. 18³=5832, 19³=6859. So between 17 and 19, there's no cube starting with 5 except 5832, which doesn't have У=1. So this case has no solution.Case 4: КУБ=729 (К=7, У=2, Б=9)Then, the 4-digit number is 7Р2Г. Digits Р and Г must not be 7,2,9. Available digits: 0,1,3,4,5,6,8. Now, 7Р2Г is a 4-digit cube starting with 7. The cube roots of numbers from 7000 to 7999 are between approximately 19.1 and 21.5. So possible cube roots: 19, 20, 21.19³=6859: starts with 6, invalid.20³=8000: starts with 8, invalid.21³=9261: starts with 9, invalid.Wait, that seems like there's no cube between 7000 and 7999? Wait, that can't be. Wait, 19³=6859, 20³=8000. So there is no cube between 7000 and 7999? That seems correct. 19³ is 6859, 20³ is 8000. So there's a gap between 6859 and 8000. Therefore, there are no 4-digit cubes starting with 7. Therefore, this case also has no solution.Therefore, from all cases, the only valid solutions are:1. КРУГ=1728 (12³), КУБ=125 (5³)2. КРУГ=2197 (13³), КУБ=216 (6³)Wait, let me verify again.First solution:КУБ=125 (5³): К=1, У=2, Б=5КРУГ=1728 (12³): digits 1,7,2,8. All unique, no overlap with Б=5. Correct.Second solution:КУБ=216 (6³): К=2, У=1, Б=6КРУГ=2197 (13³): digits 2,1,9,7. All unique, no overlap with Б=6. Correct.Third case with КУБ=512: no solution. Fourth case with 729: no solution.Therefore, the possible values are 1728 & 125, and 2197 & 216.Wait, but let me check if there are other 4-digit cubes that I might have missed.For example, in the first case where К=1, we considered 12³=1728, but are there other cubes starting with 1?10³=1000 (invalid due to repeating digits and У=0 instead of 2)11³=1331 (repeats digits 1 and 3, also У=3 instead of 2)12³=1728 (valid)13³=2197 (starts with 2, which is another case)14³=2744 (starts with 2, digits repeat 4)15³=3375 (starts with 3)16³=4096 (starts with 4)17³=4913 (starts with 4)18³=5832 (starts with 5)19³=6859 (starts with 6)20³=8000 (starts with 8)21³=9261 (starts with 9)So indeed, the only cubes starting with 1 are 1000, 1331, 1728. Only 1728 is valid for the first case. Similarly, starting with 2, we have 2197, 2744, etc. 2197 is valid for the second case.Therefore, these are the only two solutions.Final AnswerThe possible values are КРУГ = boxed{1728} (which is (12^3)) and КУБ = boxed{125} (which is (5^3)), as well as КРУГ = boxed{2197} (which is (13^3)) and КУБ = boxed{216} (which is (6^3)).</think>To solve the problem, we need to find all possible values of the numbers КРУГ (a 4-digit cube) and КУБ (a 3-digit cube) where each letter represents a unique digit.First, we list all 3-digit cubes:- (5^3 = 125)- (6^3 = 216)- (7^3 = 343) (discarded because К and Б would be the same)- (8^3 = 512)- (9^3 = 729)We analyze each valid 3-digit cube to see if we can form a corresponding 4-digit cube КРУГ with the same digits К and У, and distinct digits for Р, Г, and Б.1. КУБ = 125 (К=1, У=2, Б=5) - The 4-digit number КРУГ must start with 1 and have У=2. - Possible cube: (12^3 = 1728) - Digits: 1, 7, 2, 8 (all unique and distinct from 5) - Solution: КРУГ = 1728, КУБ = 1252. КУБ = 216 (К=2, У=1, Б=6) - The 4-digit number КРУГ must start with 2 and have У=1. - Possible cube: (13^3 = 2197) - Digits: 2, 1, 9, 7 (all unique and distinct from 6) - Solution: КРУГ = 2197, КУБ = 2163. КУБ = 512 (К=5, У=1, Б=2) - The 4-digit number КРУГ must start with 5 and have У=1. - No valid cubes found in the range 5000-5999 with the required digits.4. КУБ = 729 (К=7, У=2, Б=9) - The 4-digit number КРУГ must start with 7 and have У=2. - No valid cubes found in the range 7000-7999.Thus, the valid solutions are:- КРУГ = 1728 ((12^3)) and КУБ = 125 ((5^3))- КРУГ = 2197 ((13^3)) and КУБ = 216 ((6^3)) Final AnswerThe possible values are КРУГ = boxed{1728} (which is (12^3)) and КУБ = boxed{125} (which is (5^3)), as well as КРУГ = boxed{2197} (which is (13^3)) and КУБ = boxed{216} (which is (6^3)).]

🔑:Answer: 1728 and 125.Solution: Notice that in the words КУБ (CUBE) and КРУГ (CIRCLE), two digits repeat - К and У.Let's list all possible three-digit numbers that are cubes of some numbers:5^{3}=125, 6^{3}=216, 7^{3}=343, 8^{3}=512, 9^{3}=729 (4^3=64 - two-digit; 10^3=1000 - four-digit). Notice that 343 does not work (K ≠ B).Let's list the possible values for letters К and У:| mathbf{y} | 1 | 2 || :--- | :--- | :--- || mathbf{K} | 2 or 5 | 1 or 7 |Let's list all possible four-digit numbers that are cubes of some numbers: 10^{3}=1000, 11^{3}=1331, 12^{3}=1728, 13^{3}=2197, 14^{3}=2744, 15^{3}=3375, 16^{3}=4096, 17^{3}=4913, 18^{3}=5832, 19^{3}=6859, 20^{3}=8000, 21^{3}=9261 (9^3=729 - three-digit; 22^3=10648 - five-digit).Notice that 1000, 1331, 2744, 3375, and 8000 do not work, as K, Р, У, and Г are different digits. Among the remaining options, let's list all possible values for letters К and У:| mathbf{y} | 1 | 2 | 3 | 5 | 6 | 9 || :--- | :--- | :--- | :--- | :--- | :--- | :--- || mathbf{K} | 4 | 1 | 5 | 6 | 9 | 2 or 4 |Notice that К and У in the numbers КРУГ and КУБ match only when У = 2, К = 1. Therefore, there is a unique solution: КРУГ = 1728, КУБ = 125.## Criteria:7 points - correct answer and complete correct justification;4-5 points - answer found correctly, but the solution is insufficiently justified;3 points - the idea to consider the same letters У and К, but the solution is incomplete or has errors;Only answer - 0 points.

🧠:[<think>Alright, so I need to find the smallest natural number that has exactly 55 natural divisors. Hmm, okay. Let me think about how numbers and their divisors work. I remember that the number of divisors a number has is related to its prime factorization. Let me try to recall that formula.If a number N can be expressed as the product of primes raised to their respective exponents, like N = p₁^a₁ × p₂^a₂ × ... × pₙ^aₙ, then the total number of divisors of N is (a₁ + 1)(a₂ + 1)...(aₙ + 1). Right? Because each exponent can range from 0 up to aᵢ when building a divisor. So for each prime, you have (exponent + 1) choices, and multiplying those together gives the total number of divisors.So the problem is now to find the smallest N such that the product of (a₁ + 1)(a₂ + 1)...(aₙ + 1) equals 55. And N should be as small as possible. So I need to find exponents a₁, a₂, ..., aₙ such that when each is increased by 1 and multiplied, the result is 55, and then compute p₁^a₁ × p₂^a₂ × ... × pₙ^aₙ for the smallest primes possible, ordered by size.First, let's factorize 55 to see possible combinations of exponents. 55 is 5 × 11, which is a prime number times another prime number. Also, 55 is 55 × 1. So the possible ways to express 55 as a product of integers greater than 1 are: 55 × 1, 11 × 5. Since 55 itself is not prime, but 5 and 11 are primes.Therefore, the exponents plus one would have to multiply to 55. So exponents are one less than those factors. So possible exponent combinations:Case 1: Single exponent: 55 - 1 = 54. So N = p₁^54. Then the number would be 2^54, which is a huge number. Probably not the smallest.Case 2: Two exponents: 11 - 1 = 10 and 5 - 1 = 4. So exponents 10 and 4. Then N would be p₁^10 × p₂^4. Alternatively, since 5 comes before 11 in the factors, maybe exponents 4 and 10. But which order is better? Since primes are ordered, using higher exponents on smaller primes can lead to a smaller number. Let me check.If we take exponents 10 and 4, then assign the larger exponent to the smaller prime. So 2^10 × 3^4. Let's compute that: 2^10 is 1024, 3^4 is 81. Multiplying them: 1024 × 81. Hmm, 1024 × 80 = 81920, plus 1024 = 82944. Alternatively, if we do 2^4 × 3^10. 2^4 is 16, 3^10 is 59049. Multiplying: 16 × 59049 = 944784. So 82944 is smaller than 944784, so better to put the higher exponent on the smaller prime. So 2^10 × 3^4 is better.Case 3: Alternatively, 55 can be broken down into more factors if possible, but since 55 is 5 × 11 and both are primes, we can't break it down further. So the only possible factorizations into integers greater than 1 are 55 and 5 × 11. So the exponent combinations would be either a single exponent of 54 (which gives a very large number) or two exponents of 10 and 4 (since 10 + 1 = 11, 4 + 1 = 5). Are there any other factorizations?Wait, 55 is 55, which is 5 × 11. Are there other ways to write it as a product? For example, if we use 1s, but since in the formula each (a_i + 1) has to be at least 2 (since exponents are at least 1, right? Wait, no, exponents can be 0, but in the prime factorization, exponents are at least 1. Wait, no. Wait, when we factorize N into primes, each exponent is at least 1. Therefore, each (a_i + 1) is at least 2. Therefore, the factors of 55 must all be at least 2, so the only factorizations are 55 and 5 × 11. So no, there are no other possibilities.Therefore, the possible exponent structures are either [54] or [10, 4]. As we saw, the single exponent gives 2^54, which is enormous, and the two exponents give 2^10 × 3^4 = 82944 or similar. So 82944 is much smaller. But wait, maybe there's another way? Let me check.Wait, maybe 55 can also be expressed as a product of more numbers if we consider 1s. For example, 55 = 5 × 11 × 1. But since in the divisor formula, each (a_i + 1) must be at least 2, because exponents a_i are at least 1. Therefore, 1s are not allowed. So the factorization of 55 must be into factors each at least 2. So the only factorizations are 55 and 5 × 11, as before.So the two cases are [54] and [10,4]. Then between these, the two-exponent case is better. So N would be 2^10 × 3^4, which is 1024 × 81 = 82944. Is there a smaller number with exactly 55 divisors?Wait, let me verify. Let's check if 82944 is indeed the smallest. Let's see if there are other exponent combinations. Wait, maybe if we have three exponents? Let me think. But 55 factors into 5 × 11, so if we try to split 55 into three factors, each at least 2, but 5 and 11 are primes. So 55 can't be split into three factors each at least 2, except 5, 11, 1, but 1 is invalid. Therefore, no. Similarly, 55 is not divisible by 3, so you can't split into three factors. So I think that's not possible.Wait, unless we use exponents where some (a_i + 1) is 1, but that would mean a_i = 0, which would correspond to primes not present in the factorization. But since in the prime factorization, we can have primes with exponent 0? Wait, no. The prime factorization only includes primes that are actually present with exponent at least 1. So exponents can't be zero in the prime factorization. So all factors in the divisor count formula must be at least 2, so the factors of 55 must all be at least 2. So no, there's no way to split 55 into more than two factors each at least 2. Therefore, the only possibilities are one or two exponents.Therefore, between the two, the two exponents case is better, leading to N = 2^10 × 3^4 = 82944. But wait, let me check if another combination could give a smaller N. For example, if the exponents are arranged differently, but since we have exponents 10 and 4, assigned to primes 2 and 3, maybe another order. Wait, if we take 2^4 × 3^10, that gives 16 × 59049 = 944784, which is bigger. So 82944 is better.Alternatively, perhaps using more primes with exponents. Wait, but how? Since 55 can't be expressed as a product with more factors. Unless there is a different prime factorization for 55? Wait, 55 is 5 × 11, so if we have three primes, but how?Wait, perhaps I made a mistake here. Let me reconsider. The number of divisors is (a₁ + 1)(a₂ + 1)...(aₙ + 1) = 55. So if I can write 55 as a product of more factors, even if those factors are not prime. For example, 55 can be written as 55 × 1 × 1, but those 1s correspond to exponents of 0, which would mean primes that aren't actually in the factorization. So if I have, say, (54 + 1)(0 + 1)(0 + 1)... but since primes with exponent 0 are not included, then only the first factor counts. So no, that's equivalent to the single exponent case.Alternatively, if I can break 55 into smaller factors, even if they aren't prime. For example, 55 = 5 × 11, but 5 and 11 are primes. Wait, but 55 is 5 × 11, which are primes. So even if I try to split further, but since 5 and 11 are primes, I can't. So the number of exponent terms can't be more than two. Therefore, the minimal number should be when we have exponents 10 and 4 assigned to the smallest primes. So 2^10 × 3^4.But wait, let me think again. Suppose that instead of taking the product of two factors (5 and 11), we take three factors but one of them is 1. Wait, but as before, (a_i + 1) can't be 1 because that would require a_i = 0, which is not allowed in the prime factorization. So, no. So the only valid factorizations are 55 and 5 × 11.Therefore, the two possible exponent tuples are [54] and [10,4]. Assigning these exponents to the smallest primes. Since 2^10 × 3^4 is 82944, and 3^10 × 2^4 is 944784, which is much larger, so 82944 is better.But is 82944 the smallest? Let me check if there's a number smaller than that with exactly 55 divisors. Let's see. Maybe using different exponents. Wait, but how?Wait, another thought. The total number of divisors is 55. 55 is also equal to (4 + 1)(10 + 1), but also, since multiplication is commutative, but we've already considered that. Alternatively, is there a way to write 55 as (6 + 1)(6 + 1 + ...), but 55 divided by 7 is approximately 7.85, which is not integer. So no. Let me check all possible factorizations.55 factors are 1, 5, 11, 55. So possible pairs: (1,55), (5,11). As discussed, we can't use 1. So only (5,11). Therefore, the exponents must be 4 and 10. So the only possibilities.Therefore, the minimal number is 2^10 × 3^4. Let me compute that again to confirm. 2^10 is 1024, 3^4 is 81. 1024 × 81. Let's compute 1000 × 81 = 81,000, and 24 × 81 = 1,944. Adding them together: 81,000 + 1,944 = 82,944. So yes, 82,944.But let me check if there is a number smaller than 82,944 that has exactly 55 divisors. For example, numbers with exponents assigned to different primes. Wait, 2^4 × 3^10 is much bigger, as we saw. What about using three primes? Wait, but how? Since 55 can't be expressed as a product of three numbers greater than 1. Unless, perhaps, exponents with (a+1)(b+1)(c+1) = 55. But since 55 factors into 5 × 11, and both are primes, we can't split into three factors. So even if we try, we would need to include 1s, which aren't allowed. So that's not possible. Therefore, no.Alternatively, maybe there's a mistake in my initial assumption. Let me think again. Wait, the formula for the number of divisors is the product of (exponent + 1). So if a number has a prime factorization with exponents a, b, c,..., then the number of divisors is (a+1)(b+1)(c+1)... So if we can write 55 as a product of more than two factors, even if we have to use exponents 0, but since exponents can't be 0 in the prime factorization. Wait, but exponents in the prime factorization can't be zero. So for example, if we have a number like 2^a * 3^b * 5^c, then each of a, b, c is at least 1, so (a+1), (b+1), (c+1) are at least 2.But 55 can't be written as a product of three numbers each at least 2, since 55 = 5 × 11. So 5 and 11 are primes, so you can't break them down further. So even if we try to use three primes, we can't. For example, (2+1)(2+1)(something) but 3 × 3 × something = 55. But 3 × 3 = 9, and 55 divided by 9 is about 6.11, which is not an integer. So that's impossible.Alternatively, (4+1)(2+1)(something). 5 × 3 × something = 55. 5 × 3 = 15, 55 / 15 ≈ 3.666, not integer. Similarly, (6+1)(something). 7 × 8 = 56, which is more than 55, so no.Therefore, it's impossible to have three factors. Therefore, the number must be either of the form p^54 or p^10 q^4, where p and q are primes.Since p^54 is way too big, the smallest possible number is p^10 q^4 with the smallest primes.Now, 2^10 * 3^4 = 1024 * 81 = 82,944. Let me check if that's correct.But hold on, maybe another combination with more primes but exponents arranged in a way that the overall product is smaller. Wait, but we can't. Because we can't express 55 as a product of more than two factors. So even if we try to use three primes, we can't get the exponents such that the product of (a+1)(b+1)(c+1) is 55, since 55 can't be divided into three integers each at least 2.Therefore, the minimal number is 2^10 * 3^4. Let me check if there's a number smaller than this which has 55 divisors. Let me think about the structure of numbers with exactly 55 divisors.Since 55 is 5 × 11, any number with 55 divisors must have a prime factorization where the exponents are 4 and 10 (since 4 + 1 = 5 and 10 + 1 = 11). Therefore, the exponents have to be 4 and 10, assigned to two different primes. To get the minimal number, we should assign the larger exponent to the smaller prime, right? Because 2^10 is smaller than 3^10, and 3^4 is smaller than 5^4. So 2^10 * 3^4 is the minimal combination.Alternatively, if we take 2^4 * 3^10, that would be 16 * 59049 = 944,784, which is way larger. Similarly, using higher primes like 5^10 would be even worse. So 2^10 * 3^4 is the minimal.Wait, but let's check another thing. If the exponents are arranged as [10,4], then using the primes 2 and 3. But what if we use more primes with exponents, but with exponents that multiply to 55 when each is increased by 1? Wait, but 55 can't be factored into more than two numbers greater than 1. Therefore, that's not possible.Alternatively, maybe using exponents such that one exponent is 54, but that's 2^54, which is enormous, so it's not going to be smaller.Therefore, the conclusion is that the smallest number with exactly 55 divisors is 2^10 * 3^4 = 82944.But let me confirm with an example. Let's check the number 60, which is 2^2 * 3^1 * 5^1. The number of divisors is (2+1)(1+1)(1+1) = 3*2*2=12. So that's correct.Alternatively, take 2^4 * 3^4 = 16 * 81 = 1296. The number of divisors is (4+1)(4+1)=25. Not 55.If we take 2^10 * 3^4, the number of divisors is (10+1)(4+1)=11*5=55. Correct.Alternatively, check 2^10 * 3^4: 2^10 is 1024, 3^4 is 81. 1024*81=82944.Is there a number smaller than 82944 with 55 divisors? Let's try to see.Suppose we take another combination, like 2^something * 3^something * 5^something, but since we can't get 55 divisors with three primes, as we saw, it's impossible. So the minimal should be 82944.But let me check the number 25200. Wait, 25200 is a number with a lot of divisors. Let me check how many divisors it has. 25200's prime factorization is 2^4 * 3^2 * 5^2 * 7^1. So number of divisors is (4+1)(2+1)(2+1)(1+1)=5*3*3*2=90. So 90 divisors, which is more than 55. So not helpful.How about 10080? 10080's prime factors: 2^5 * 3^2 * 5^1 * 7^1. Number of divisors: (5+1)(2+1)(1+1)(1+1)=6*3*2*2=72. Still more than 55.How about 5040? 5040 = 2^4 * 3^2 * 5^1 * 7^1. Number of divisors: (4+1)(2+1)(1+1)(1+1)=5*3*2*2=60. Close, but 60 divisors. Still not 55.Hmm, 60 divisors. Let's see 55 is between 48 and 60. Let me check 4200. 4200 is 2^3 * 3^1 * 5^2 * 7^1. Divisors: (3+1)(1+1)(2+1)(1+1)=4*2*3*2=48. So 48.So between 4200 (48 divisors) and 5040 (60 divisors), maybe the number we found, 82944, is in that range. But 82944 is 82,944, which is much larger than 5040. Wait, but perhaps there's a smaller number with exactly 55 divisors. Wait, 55 is a less common number of divisors. Let me check smaller numbers.Wait, maybe 2^10 * 3^4 is indeed the smallest. Let's see. Let's try to think of numbers with two prime factors. For example, numbers like 2^4 * 3^10, which is 16 * 59049 = 944,784. So that's way bigger. If we take other primes, like 2^10 * 5^4, that's 1024 * 625 = 640,000, which is also bigger. So 2^10 * 3^4 is smaller.Alternatively, 3^10 * 2^4, which we already saw. So 82,944 seems to be the smallest possible.Wait, but let me check numbers with three prime factors. Even though the divisor count formula requires 55 = (a+1)(b+1)(c+1)... but we saw that 55 cannot be broken down into three factors each at least 2. Since 55 = 5*11, which are primes, so even if you try to split 55 into three factors, one of them has to be 1, which is invalid. Therefore, numbers with three prime factors would have exponents such that (a+1)(b+1)(c+1) = 55, but since we can't have 1s, that's impossible.Therefore, numbers with three prime factors can't have exactly 55 divisors. Therefore, the number must have either one or two prime factors. The one prime factor case is too big, so the two prime factors case is the way to go.Therefore, 2^10 * 3^4 = 82,944 is the smallest number with exactly 55 divisors.Wait, but let me confirm with another example. Let's check 60. 60 has prime factorization 2^2 * 3^1 * 5^1, so number of divisors is (2+1)(1+1)(1+1) = 12, which is correct.Another example: 2^4 * 3^2 = 16*9=144. Number of divisors is (4+1)(2+1)=15.If I take 2^10 * 3^4, which is 1024*81=82,944. Let's check the number of divisors: (10+1)(4+1)=11*5=55. Correct.Is there a smaller number? Let's check 2^10 is 1024, 3^4 is 81. Maybe a composite number between, say, 10,000 and 82,944 that has 55 divisors. Let's see.Alternatively, let's check 30,240. Wait, 30,240 is 2^5 * 3^3 * 5^1 * 7^1. Number of divisors: (5+1)(3+1)(1+1)(1+1)=6*4*2*2=96. No.How about 20,160: 2^6 * 3^2 * 5^1 * 7^1. Divisors: (6+1)(2+1)(1+1)(1+1)=7*3*2*2=84.Still higher. 16, which is 16, but 16 has 5 divisors. Not helpful.Wait, maybe 6048. 6048 is 2^5 * 3^3 * 7^1. Number of divisors: (5+1)(3+1)(1+1)=6*4*2=48.Still not 55. Hmm.Alternatively, let's take 2^4 * 3^4 = 16*81=1296. Number of divisors: (4+1)(4+1)=25. Still lower than 55.What about 2^6 * 3^4 = 64*81=5184. Divisors: (6+1)(4+1)=7*5=35.Still lower. 2^8 * 3^4 = 256*81=20736. Divisors: (8+1)(4+1)=9*5=45.Still lower. 2^10 *3^4=82944. Divisors: 11*5=55.Okay, so between 20736 (45 divisors) and 82944 (55 divisors), maybe there's no number in between with exactly 55 divisors. Because if you try to add another prime factor, you need to split the exponent product into 55, which requires two exponents. But since 55 can't be broken down into more factors, adding another prime would require exponents that would actually increase the number's size.For example, suppose we take 2^4 *3^2 *5^10. That would be 16*9*9765625= which is a huge number. So even if we use a high exponent on a larger prime, the number becomes enormous.Alternatively, 2^a *3^b *5^c where (a+1)(b+1)(c+1)=55. But since 55 is 5*11, it's not possible to split into three factors each at least 2. Therefore, such a number can't exist with three primes.Therefore, the minimal number is indeed 82944.But let me check the OEIS sequence for numbers with exactly 55 divisors. Wait, I don't have access to that right now, but perhaps I can reason it out.Alternatively, think of the general method for finding the smallest number with a given number of divisors. The standard approach is to generate all possible exponent combinations that multiply to the desired divisor count, sort the exponents in decreasing order, assign the largest exponents to the smallest primes, and compute the product. Then choose the smallest such product.In this case, desired divisor count is 55=5×11. The exponents are one less: 4 and 10. Sort exponents in decreasing order: [10,4]. Assign to primes [2,3]. So 2^10 ×3^4=1024×81=82944.Alternatively, if the exponents were [4,10], assign to [2,3], which gives 2^4 ×3^10=16×59049=944784. So the former is smaller.Another way: If the exponents are sorted in descending order and assigned to the primes in ascending order, we get the minimal number. So yes, 10 and 4 sorted descending is [10,4], assigned to 2 and 3, so 2^10×3^4=82944.Therefore, this should be the minimal number.But just to ensure, let's check if there is a smaller number by using exponents in a different way. For example, using 55= (54+1), leading to a single prime factor. 2^54 is way too big, so no. Also, other primes like 3^54 would be even larger.Alternatively, a number like 2^4 *3^4 *5^4 *7^4... but that would have (4+1)^k divisors, which for k=2 is 25, k=3 is 125, so no.Alternatively, mixing exponents. But since 55 is not a power, but a product of two primes, you can't get it by multiplying multiple equal factors. So no.Therefore, I think the answer is 82944. So the smallest natural number with exactly 55 divisors is 82,944.Final AnswerThe smallest natural number with exactly 55 natural divisors is boxed{82944}.</think>To find the smallest natural number with exactly 55 natural divisors, we start by noting that the number of divisors of a number ( N ) with prime factorization ( N = p_1^{a_1} times p_2^{a_2} times cdots times p_n^{a_n} ) is given by ( (a_1 + 1)(a_2 + 1) cdots (a_n + 1) ). We need this product to equal 55.First, we factorize 55:[ 55 = 5 times 11 ]This means the exponents in the prime factorization of the number must satisfy:[ (a_1 + 1)(a_2 + 1) = 55 ]The possible pairs of exponents (considering the factors 5 and 11) are:[ (a_1, a_2) = (4, 10) ]We need to assign these exponents to the smallest primes to minimize the number. The smallest primes are 2 and 3. Assigning the larger exponent to the smaller prime gives a smaller number:[ 2^{10} times 3^4 ]Calculating this value:[ 2^{10} = 1024 ][ 3^4 = 81 ][ 1024 times 81 = 82944 ]We verify that this number has exactly 55 divisors:[ (10 + 1)(4 + 1) = 11 times 5 = 55 ]Thus, the smallest natural number with exactly 55 natural divisors is (boxed{82944}).,<think>Okay, so I need to find the smallest natural number that has exactly 55 natural divisors. Hmm, let me think. I remember that the number of divisors of a number relates to its prime factorization. Right, if a number N can be written as a product of primes raised to their respective exponents, like N = p₁^a₁ * p₂^a₂ * ... * pₙ^aₙ, then the total number of divisors is (a₁ + 1)*(a₂ + 1)*...*(aₙ + 1). So, for example, if a number is 2^3 * 3^2, the number of divisors would be (3+1)*(2+1) = 4*3 = 12. So, applying that here, I need to find a number such that when you take the exponents in its prime factorization, add one to each, and multiply them together, you get 55. And among all such numbers, I need the smallest one. Got it. First, let me factorize 55 to see how it can be expressed as a product of integers greater than 1. 55 is 5 * 11. Since 5 and 11 are both prime numbers, those are the only factors. Wait, but 55 can also be written as 55 itself. So, the possible ways to express 55 as a product of integers greater than 1 are:1. 55 = 552. 55 = 5 * 11Are there any other factors? Let's check. 55 divided by 1 is 55, 5, 11, and that's it. So yes, only those two factorizations. Now, each of these corresponds to possible exponents in the prime factorization of the number. Because the number of divisors is (a₁ + 1)*(a₂ + 1)*... So, if the total is 55, then the exponents plus one should multiply to 55. So, for the first case, 55 = 55. That would mean that there's a single exponent plus one equal to 55. So, exponent would be 54. So, the number would be a prime number raised to the 54th power. But that's going to be a huge number, right? Like 2^54, which is already a very large number.Second case, 55 = 5 * 11. That would mean that there are two exponents, one plus one is 5, so exponent is 4, and the other plus one is 11, so exponent is 10. Alternatively, the other way around. So, the exponents are 4 and 10. Then, the number would be the product of two primes raised to 10 and 4. But which primes? To get the smallest number, we should use the smallest primes, right? Since smaller primes raised to higher powers will contribute less to the overall size of the number. For example, 2^10 * 3^4 versus 2^4 * 3^10. Let's compute both.First, 2^10 is 1024, and 3^4 is 81, so 1024 * 81 = let's see. 1000*81 = 81,000 and 24*81 = 1,944. So total is 82,944. On the other hand, 2^4 is 16, 3^10 is 59,049. So 16 * 59,049 = 944,784. So, definitely, 2^10 * 3^4 is much smaller. So, the ordering of exponents matters. So, to minimize the number, we should assign the higher exponents to the smaller primes. So, higher exponents on smaller primes. So, 10 on 2 and 4 on 3 would be better. Wait, but 10 is higher than 4. Wait, but in the first case, it's 2^10 * 3^4, and in the second case, 2^4 * 3^10, which is bigger. So, yeah, higher exponents on smaller primes gives a smaller number. Makes sense.Alternatively, could there be more factors? Wait, but 55 is 5*11, which are primes, so we can't break it down further. So, the possible exponents configurations are either a single exponent of 54, or two exponents of 4 and 10. Because (a+1)(b+1) = 55, so possible (54) or (4,10). Are there any other possibilities? Let me check.If we consider 55 as a product of more numbers, like three numbers? But since 55 is 5*11, which are primes, we can't express it as a product of three numbers greater than 1. For example, 5*11*1, but 1 is not allowed here because when we talk about exponents, adding 1, so each factor must be at least 2. So, no, 55 can only be written as 55 or 5*11. Therefore, only two possible configurations.So, between these two, the single exponent 54 gives a number of 2^54, which is enormous, while the two-exponent case gives 2^10 * 3^4 = 82,944. Which is way smaller. So, 82,944 is a candidate.But wait, is that the minimal? Let me check. Wait, perhaps there are other factorizations. Let me think again. Wait, 55 is 5*11, but in terms of exponents, does order matter? For example, exponents in the prime factorization can be arranged in any order, but since primes are in order, to get the minimal number, we need to assign higher exponents to smaller primes. So, as we did before.But hold on, is there another way to express 55 as a product? Let me see. Wait, 55 is also 55*1, but 1 is not helpful here because exponents correspond to (a+1), so (a+1)=55 and (a+1)=1. But (a+1)=1 would mean exponent 0, which is just not having that prime. So, that's the same as the first case.Alternatively, if we consider 55 as 11*5, but that's the same as 5*11. So, same as before.So, only two configurations. So, 82,944 is the candidate. But is there a smaller number with exactly 55 divisors? Let me check.Wait, but perhaps there's a way to get 55 as a product of more factors if we allow exponents to have +1s that include 1. Wait, but 55 is 5*11, which are primes, so if we want to write it as a product of more numbers, we would need to use 1s, but exponents can't be negative. So, for example, 55 = 5*11*1, but the 1 would correspond to an exponent of 0, which would mean another prime factor raised to 0, which is 1. So, that would not contribute. So, in effect, it's the same as two exponents. So, for example, if we had exponents 4, 10, and 0, then the number would be p1^4 * p2^10 * p3^0 = p1^4 * p2^10, which is the same as the two-exponent case. So, that doesn't help.Therefore, only two possible configurations. So, the minimal number is either 2^54 or 2^10 * 3^4. But 2^10 * 3^4 is much smaller, so that's the answer. Wait, but let me check.Wait, 2^10 is 1024, 3^4 is 81, 1024*81=82,944. Alternatively, 2^4 * 3^10 is 16 * 59,049=944,784, which is way bigger. So, 82,944 is smaller. But is there a number with three prime factors? Let me see.Wait, if we tried to express 55 as a product of three numbers, but as we saw, 55 cannot be expressed as a product of three numbers greater than 1. But perhaps with exponents? Wait, but exponents plus 1 have to multiply to 55. So, if I have three exponents, then (a+1)(b+1)(c+1)=55. But since 55 is 5*11, which can't be split into three numbers greater than 1, unless we use 1s. So, for example, (5*11*1), but that would correspond to exponents 4,10,0. But the prime with exponent 0 would not be part of the number, so effectively, it's the same as two primes. So, no, that doesn't help.Alternatively, maybe using more primes with exponents? But since 55 is 5*11, and we can't split into more factors, we can't have more exponents. So, the only possibilities are two primes with exponents 4 and 10, or one prime with exponent 54.Therefore, the minimal number should be 2^10 * 3^4. But let's check if another combination with more primes can result in a smaller number. Wait, but how?Wait, if we use more primes, but exponents plus 1 must multiply to 55. Since 55 is 5*11, which are primes, we can't have more than two factors. Unless we use exponents plus 1 equal to 1, but that would require exponents to be 0, which again, are not contributing.Wait, unless we have exponents arranged differently. Wait, but 55 is 5*11. So, if we have three primes, then we would need (a+1)(b+1)(c+1)=55. But since 55 can't be factored into three numbers greater than 1, the only way is 5*11*1, which as before, gives exponents 4,10,0. So, same as two primes.Alternatively, if we have exponents plus 1 as 5,11,1, which would correspond to exponents 4,10,0. So, the number would be 2^4 *3^10 *5^0=2^4 *3^10=16*59049=944,784, which is way bigger. So, that's not helpful. So, again, the minimal is 2^10 *3^4=82,944.But wait, let me confirm this. Let me check numbers with more prime factors. For example, suppose we try to use exponents such that (a+1)(b+1)(c+1)...=55, but since 55 is 5*11, we can't have more than two factors. So, unless we have exponents plus 1 as 55, or 5*11. So, even if we have more primes, they would have to have exponents of 0, which don't contribute.So, the only possibilities are either a single prime raised to 54, or two primes with exponents 4 and 10. Therefore, 2^10 *3^4 is the minimal. But let me check if 2^10 *3^4 is indeed 82,944. Let's compute:2^10 is 1024. 3^4 is 81. Then, 1024*81. Let's compute 1000*81 = 81,000. Then, 24*81 = 1,944. So, 81,000 + 1,944 = 82,944. Okay, that's correct.Is there a smaller number? For example, if we use primes 2,3,5, but with exponents that somehow multiply to 55? Wait, but we can't. Because (a+1)(b+1)(c+1) would need to be 55, but as we saw, 55 can't be split into three factors greater than 1, so we can't have three primes. Unless we have exponents 4,10,0, but that's the same as two primes.Alternatively, maybe using higher exponents on smaller primes. Wait, but we already did that. Assigning the higher exponent (10) to the smaller prime (2), and the lower exponent (4) to the next prime (3). So, 2^10 *3^4=82,944.But let me check another arrangement. Suppose we use exponents 2 and something else. Wait, 55 factors as 5*11. So, exponents plus 1 must be 5 and 11, so exponents 4 and 10. So, no, there's no way around that. Unless we use exponents plus 1 as 55, which is a single prime. So, 2^54 is 18014398509481984, which is way, way larger. So, definitely not.Therefore, 82,944 is the minimal. But wait, let me check if there is another combination where exponents can be arranged such that the number is smaller.Wait, here's an idea. Maybe using exponents 2, 5, and 5, but 2+1=3, 5+1=6, so 3*6=18, which is not 55. Hmm, that's not helpful.Wait, but maybe if the exponents are arranged differently. Wait, 55 is 5*11. So, unless there is a different way to break down 55, but since 5 and 11 are primes, we can't. So, it's not possible. So, I think 82,944 is the answer.But let me check with another set of exponents. Wait, suppose we use three primes. Wait, but (a+1)(b+1)(c+1) would need to be 55. As 55 is 5*11, and if we have three primes, then (a+1)(b+1)(c+1) = 5*11*1. So, exponents would be 4,10,0. So, the number would be 2^4*3^10*5^0 = 16*59049 = 944,784, which is larger. So, not helpful.Alternatively, if we have four primes. Then, (a+1)(b+1)(c+1)(d+1)=55. But 55 is 5*11, so factors would be 5*11*1*1, which gives exponents 4,10,0,0. So, the number would be 2^4*3^10*5^0*7^0= same as before. Still larger.Therefore, no. So, the minimal number is 2^10*3^4=82,944.Wait, but let me check if that number actually has 55 divisors. Let's verify. The prime factorization is 2^10 *3^4. So, the number of divisors is (10+1)*(4+1)=11*5=55. Yes, that's correct. So, 82,944 has 55 divisors. And since we can't get a smaller number by using more primes, because that would require higher exponents on smaller primes which would lead to a larger product, this should be the minimal number.But wait, let me check another possibility. Suppose there is a number with a different prime factorization structure. For example, using 2, 3, and 5, but with exponents such that the number of divisors is 55. Let's see. For example, if we have exponents such that (a+1)(b+1)(c+1)=55. But 55 factors into 5*11, so unless one of the exponents is 0, we can't have three primes. So, same as before. So, no.Alternatively, maybe using exponents with higher multiplicities? Wait, but the number of divisors is a multiplicative function based on the exponents. So, unless we can factor 55 into more factors, but we can't. Therefore, no.Alternatively, maybe if we use exponents 54, but with a larger base? Like 3^54, but that's even larger than 2^54, which is already too big. So, no.Therefore, 82,944 is the minimal. Wait, but let me check if there's a number smaller than 82,944 with 55 divisors. Let me think. For example, 2^10*3^4=82,944. Let me see if another combination of primes could give a smaller number.Suppose we use 2^4 *3^10=944,784, which is bigger, as before.Alternatively, 2^10 *5^4=1024*625=640,000, which is bigger than 82,944. Similarly, 2^10*7^4=1024*2401=2,459,824, which is way bigger.Alternatively, 3^10*5^4=59049*625=36,905,625, which is way bigger.So, no, using other primes gives bigger numbers.Wait, perhaps using more primes with exponents? For example, 2^a *3^b *5^c *... such that (a+1)(b+1)(c+1)...=55. But as established before, since 55=5*11, we can't have more than two primes without including exponents of 0, which don't contribute. So, the minimal is indeed two primes with exponents 10 and 4.Alternatively, perhaps using a different exponent combination. Wait, 55 is 5*11. So, if we use exponents 10 and 4, but maybe using different primes? For example, 2^4 *3^10 vs. 2^10 *3^4. But 2^10*3^4 is 82,944, while 2^4*3^10 is 944,784. So, definitely, the first one is smaller.Wait, so in order to minimize the number, we should assign the larger exponents to the smaller primes. Since the smaller primes raised to higher exponents contribute less to the overall size of the number. For example, 2^10 is 1024, which is manageable, but 3^10 is 59,049, which is way bigger. So, putting the higher exponent on the smaller prime helps keep the number smaller.Therefore, 2^10 *3^4 is the way to go.Wait, just to confirm, is there a number between, say, 1 and 82,944 that has exactly 55 divisors? Probably not, because 82,944 is already a product of the smallest primes with exponents arranged to minimize the product. But to be thorough, let's check some numbers.But checking every number up to 82,944 is impractical. However, maybe we can check numbers with different factorizations. For example, suppose a number has three prime factors. Then, the number of divisors would be (a+1)(b+1)(c+1). For example, if a number is 2^4 *3^4 *5^4, the number of divisors is (4+1)(4+1)(4+1)=125, which is more than 55. So, that's not helpful.Alternatively, if a number is 2^4 *3^2 *5^2, the number of divisors is (4+1)(2+1)(2+1)=5*3*3=45, which is less than 55. So, not enough. If we do 2^4 *3^2 *5^4, then divisors are 5*3*5=75. Still not 55.Alternatively, 2^10 *3^2. Number of divisors is (10+1)(2+1)=11*3=33. Not enough. If we have 2^6 *3^4, divisors are (6+1)(4+1)=7*5=35. Still not 55.Wait, how about 2^4 *3^4 *5^1. Then divisors are (4+1)(4+1)(1+1)=5*5*2=50. Closer, but still not 55.Or 2^4 *3^4 *5^2: (4+1)(4+1)(2+1)=5*5*3=75. Hmm.Alternatively, 2^10 *3^4: 11*5=55. Exactly 55. So, that's the one. So, unless there's a way to get 55 divisors with a different combination, but as we saw, 55 can't be factored into more than two factors unless we use 1s, which correspond to exponents of 0.So, no, there isn't a way. So, the conclusion is that 82,944 is the smallest number with exactly 55 divisors.Wait, but let me check another angle. Maybe using exponents such that (a+1)(b+1)=55, but different exponents. But 55 factors only as 1*55, 5*11. So, the exponents plus one must be 5 and 11. So, exponents 4 and 10. So, no other options. Therefore, the minimal number is indeed 2^10 *3^4=82,944.But to make sure, let's check if 82,944 is indeed the minimal. Let me compare it with another number.Suppose we have a number like 2^10 *3^4=82,944. Let's see if there's a way to rearrange the exponents among other primes to get a smaller number.For instance, if we take 2^4 *3^10, but as calculated before, that's 16*59049=944,784, which is larger. So, no.What if we use 2^10 *5^4=1024*625=640,000, which is larger. Similarly, 2^10 *7^4=1024*2401=2,459,824. So, even larger.Alternatively, 3^10 *5^4=59049*625=36,905,625. Nope. So, definitely not.Alternatively, 2^5 *3^5 *5^5. Wait, but the number of divisors would be (5+1)(5+1)(5+1)=6*6*6=216. Not 55.Alternatively, using exponents 10, 4. If we use primes 2 and 3, we get 82,944. If we use primes 2 and 5, it's 2^10*5^4=1,024*625=640,000. That's way bigger. So, no.Alternatively, using primes 3 and 2, but same as 2 and 3. So, order doesn't matter except in exponent assignment.So, the minimal number is indeed 2^10 *3^4=82,944. Therefore, the answer should be 82,944. Let me check with another method.Alternatively, perhaps I can check the list of highly composite numbers or something. But I don't have that list memorized. However, 82,944 seems reasonable.Wait, let me check smaller numbers. For example, 2520 is a highly composite number. Let's see how many divisors it has. Its prime factorization is 2^3 *3^2 *5^1 *7^1. So, number of divisors is (3+1)(2+1)(1+1)(1+1)=4*3*2*2=48. So, 48 divisors, less than 55. Next, 5040: 2^4 *3^2 *5^1 *7^1. Divisors: (4+1)(2+1)(1+1)(1+1)=5*3*2*2=60. So, 60 divisors. So, 5040 has 60 divisors. That's more than 55. So, between 2520 (48) and 5040 (60), there might be numbers with 55 divisors. But 82,944 is between them? Wait, 2520 is 2,520, 5040 is 5,040. 82,944 is 82,944, which is way larger. So, maybe there is a number between 2,520 and 82,944 with 55 divisors. But how?Wait, but according to our previous reasoning, the minimal number is 82,944. However, maybe there is a composite number between 2520 and 82,944 that has 55 divisors. Let's see. Let's try to find such a number.Suppose we take a number like 2^4 *3^4 *5^2. Then, the number of divisors would be (4+1)(4+1)(2+1)=5*5*3=75. Too many.Alternatively, 2^6 *3^4 *5^1: (6+1)(4+1)(1+1)=7*5*2=70. Still too many.2^4 *3^2 *5^2 *7^1: (4+1)(2+1)(2+1)(1+1)=5*3*3*2=90. Still too many.Wait, maybe with fewer primes. Let's see. 2^10 *3^4=82,944. But maybe there's a number like 2^4 *3^10=944,784, which is larger. Not helpful.Alternatively, 2^8 *3^6: (8+1)(6+1)=9*7=63. Still more than 55.2^7 *3^5: (7+1)(5+1)=8*6=48. Less than 55.2^9 *3^4: (9+1)(4+1)=10*5=50. Still less.2^10 *3^4=55. So, exactly 55. So, the first number that reaches 55 divisors is 82,944. Before that, numbers have fewer divisors. For example, 2^10=1024 has 11 divisors, 2^10*3=3072 has (10+1)(1+1)=22 divisors. 2^10*3^2= 1024*9=9216 has (10+1)(2+1)=33 divisors. 2^10*3^3=27,648 has (10+1)(3+1)=44 divisors. Then, 2^10*3^4=82,944 has 55. So, yeah, each time you add a factor of 3, the number of divisors increases by multiplying (3+1). Wait, no, when you go from 3^0 to 3^1, divisors go from 11 to 22. Then 3^2, 33. 3^3, 44. 3^4, 55. So, each time you add an exponent to 3, you multiply the number of divisors by (new exponent +1)/(old exponent +1). So, from 3^0 to 3^1: 11*2=22. Then 22*1.5=33, but 1.5 is not an integer. Wait, no, maybe it's (exponent +1). So, when you have 3^2, the exponent is 2, so 2+1=3. So, the number of divisors is multiplied by 3. Wait, original number is 2^10 with 11 divisors. Then, multiplying by 3^1, you get 11*(1+1)=22. Multiplying by 3^2, 11*(2+1)=33. Multiplying by 3^3, 11*(3+1)=44. Multiplying by 3^4, 11*(4+1)=55. So, that works.So, indeed, 2^10*3^4 is the first number in that sequence that reaches 55 divisors. Therefore, there are no smaller numbers with 55 divisors. Because any other number with 55 divisors would have to have a different prime factorization, which would either have a single prime with exponent 54 (which is way larger) or two primes with exponents 10 and 4. Assigning the higher exponent to the smaller prime gives the minimal number.Therefore, 82,944 is the answer. But let me check once more if there's another way. Suppose we take a number like 2^4 *3^4 *5^4, which is (2*3*5)^4=30^4=810,000. The number of divisors is (4+1)^3=125. Which is more than 55. So, not helpful.Alternatively, a number like 2^5 *3^4 *5^2: number of divisors is (5+1)(4+1)(2+1)=6*5*3=90. Still too many.Alternatively, 2^6 *3^3 *5^2: divisors=(6+1)(3+1)(2+1)=7*4*3=84. Still too many.Alternatively, 2^7 *3^3 *5^1: divisors=8*4*2=64. Still more than 55.2^4 *3^3 *5^2 *7^1: divisors=5*4*3*2=120. No.Alternatively, 2^2 *3^2 *5^2 *7^1 *11^1: divisors=3*3*3*2*2=108. No.Alternatively, 2^4 *3^2 *5^1 *7^1: divisors=5*3*2*2=60. Closer, but still 60.Wait, 60 is 60, which is more than 55. So, maybe the first number after 48 divisors is 60, then 55 is in between?But how? If the number of divisors jumps from 48 to 60, then 55 is skipped. But according to our previous calculation, 82,944 has 55. So, maybe 55 is not a common number of divisors in smaller numbers. Therefore, 82,944 is indeed the first.Therefore, after checking multiple possibilities, I can conclude that the smallest natural number with exactly 55 divisors is 82,944.Final AnswerThe smallest natural number with exactly 55 natural divisors is boxed{82944}.</think>To find the smallest natural number that has exactly 55 natural divisors, we start by noting that the number of divisors of a number ( N ) with prime factorization ( N = p_1^{a_1} cdot p_2^{a_2} cdot ldots cdot p_n^{a_n} ) is given by ( (a_1 + 1)(a_2 + 1) cdots (a_n + 1) ). We need this product to equal 55. The factorization of 55 is ( 55 = 5 times 11 ). Therefore, the exponents in the prime factorization of the number must satisfy ( (a_1 + 1)(a_2 + 1) = 55 ). This can be achieved by either:1. A single prime raised to the 54th power (since ( 54 + 1 = 55 )), which would result in a very large number.2. Two primes with exponents 4 and 10 (since ( (4 + 1)(10 + 1) = 5 times 11 = 55 )).To minimize the number, we assign the higher exponent to the smaller prime. Thus, we consider ( 2^{10} cdot 3^4 ).Calculating this value:- ( 2^{10} = 1024 )- ( 3^4 = 81 )- Multiplying these together: ( 1024 times 81 = 82944 )We verify that the number of divisors of 82944 is indeed 55:- The prime factorization is ( 2^{10} cdot 3^4 )- The number of divisors is ( (10 + 1)(4 + 1) = 11 times 5 = 55 )Thus, the smallest natural number with exactly 55 natural divisors is (boxed{82944}).]

🔑:Answer: 2^{10} cdot 3^{4}. Solution. If a natural number n has the form n=p_{1}^{k_{1}} cdot p_{2}^{k_{2}} cdot ldots cdot p_{m}^{k_{m}}, where p_{1}, p_{2}, ldots, p_{m} are its distinct prime divisors, then the number of natural divisors of n is left(k_{1}+1right)left(k_{2}+1right) ldotsleft(k_{m}+1right) (this follows from the general form of the divisor p_{1}^{i_{1}} p_{2}^{i_{2}} ldots p_{m}^{i_{m}} of the number n, 0 leq i_{1} leq k_{1}, left.0 leq i_{2} leq k_{2}, ldots, 0 leq i_{m} leq k_{m}right). Then left(k_{1}+1right)left(k_{2}+1right) ldotsleft(k_{m}+1right)=55=5 cdot 11, and therefore either m=1, or m=2. In the first case k_{1}=54 and n=p_{1}^{54}. In the second case k_{1}=4, k_{2}=10, i.e., n=p_{1}^{4} cdot p_{2}^{10}. The smallest prime numbers are 2 and 3. Therefore, it remains to compare three numbers; 2^{54}, 2^{4} cdot 3^{10} and 3^{4} cdot 2^{10}. Obviously, the smallest of them is 2^{10} cdot 3^{4}.