🧠:[<think>Okay, so the problem is asking which set of numbers cannot be the lengths of the sides of a right triangle. The options are A, B, C, and D. Each option has three numbers, and I need to check if they can form a right triangle. First, I remember that for three numbers to be the sides of a right triangle, they must satisfy the Pythagorean theorem. That is, the sum of the squares of the two shorter sides should equal the square of the longest side. The longest side is the hypotenuse. So, my plan is to check each option by squaring each number and seeing if the Pythagorean theorem holds.Let me start with option A: √3, √4, √5. Hmm, √4 is 2, right? So the sides are √3, 2, and √5. Let's square them. (√3)² is 3, (2)² is 4, and (√5)² is 5. Now, the two smaller squares are 3 and 4. Adding them together: 3 + 4 = 7. The largest square is 5. But 7 is not equal to 5, so that means option A does not satisfy the Pythagorean theorem. Hmm, so maybe A is the answer? Wait, but let me check the other options to be sure.Option B: 3, 4, 5. Classic Pythagorean triple. Let's square them: 9, 16, 25. The sum of the two smaller squares: 9 + 16 = 25, which is equal to the largest square. So B is a right triangle. That's fine.Option C: 0.3, 0.4, 0.5. Let me square these. 0.3 squared is 0.09, 0.4 squared is 0.16, and 0.5 squared is 0.25. Adding the two smaller ones: 0.09 + 0.16 = 0.25. Which is equal to 0.25. So that works. So C is a right triangle.Option D: 30, 40, 50. Squaring these: 900, 1600, 2500. Adding the two smaller squares: 900 + 1600 = 2500. Which equals the largest square. So D is also a right triangle.Wait, so all options except A satisfy the Pythagorean theorem. So that means the answer is A. But just to make sure I didn't make a mistake with A. Let's go over it again.Option A: sqrt(3), sqrt(4), sqrt(5). So sqrt(4) is 2, so the sides are approximately 1.732, 2, and 2.236. The hypotenuse should be the largest, which is sqrt(5). Squaring them gives 3, 4, 5. Wait, hold on. Wait, if you square each term, you get 3, 4, 5. So sqrt(3)^2 + sqrt(4)^2 is 3 + 4 = 7, which is not equal to sqrt(5)^2 which is 5. So yes, that's 7 ≠ 5. So definitely, A doesn't work. So A is the answer.But just to make sure, maybe there's a different order? Like, what if the hypotenuse wasn't the last one? But in the options, they probably listed them in order. But even if they weren't, the hypotenuse has to be the longest side. So in A, sqrt(5) is the largest. So the other two sides are sqrt(3) and sqrt(4). So the sum of their squares is 3 + 4 = 7, which isn't equal to 5. So yeah, no way that works. Therefore, the answer is option A.Final Answerboxed{A}</think>To determine which set of numbers cannot be the lengths of the sides of a right triangle, we check each option using the Pythagorean theorem, which states that the sum of the squares of the two shorter sides must equal the square of the longest side.Option A: (sqrt{3}), (sqrt{4}), (sqrt{5})- Squaring each term: ((sqrt{3})^2 = 3), ((sqrt{4})^2 = 4), ((sqrt{5})^2 = 5)- Sum of the squares of the two shorter sides: (3 + 4 = 7)- Square of the longest side: (5)- Since (7 neq 5), these cannot form a right triangle.Option B: 3, 4, 5- Squaring each term: (3^2 = 9), (4^2 = 16), (5^2 = 25)- Sum of the squares of the two shorter sides: (9 + 16 = 25)- Square of the longest side: (25)- Since (25 = 25), these can form a right triangle.Option C: 0.3, 0.4, 0.5- Squaring each term: (0.3^2 = 0.09), (0.4^2 = 0.16), (0.5^2 = 0.25)- Sum of the squares of the two shorter sides: (0.09 + 0.16 = 0.25)- Square of the longest side: (0.25)- Since (0.25 = 0.25), these can form a right triangle.Option D: 30, 40, 50- Squaring each term: (30^2 = 900), (40^2 = 1600), (50^2 = 2500)- Sum of the squares of the two shorter sides: (900 + 1600 = 2500)- Square of the longest side: (2500)- Since (2500 = 2500), these can form a right triangle.Thus, the set of numbers that cannot be the lengths of the sides of a right triangle is (boxed{A}).,<think>Okay, let's see. The question is asking which set of numbers cannot be the lengths of the sides of a right triangle. There are four options: A, B, C, D. I need to check each one and see if they satisfy the Pythagorean theorem, right? Remember, in a right triangle, the sum of the squares of the two shorter sides should equal the square of the longest side, which is the hypotenuse.Starting with option B: 3, 4, 5. I think that's a classic right triangle. Let me verify. The two shorter sides are 3 and 4. Squaring them: 3² + 4² = 9 + 16 = 25. The hypotenuse is 5, and 5² is 25. So yes, that works. So B is a right triangle.Option D: 30, 40, 50. Hmm, those numbers look like multiples of 3, 4, 5. If we multiply each by 10, we get 30, 40, 50. So maybe this is similar to B, just scaled up. Let me check. 30² + 40² = 900 + 1600 = 2500. And 50² is 2500. Yep, that's also a right triangle. So D is okay.Option C: 0.3, 0.4, 0.5. Those are decimals. Maybe they are scaled down versions of 3, 4, 5? Let's see. If we multiply each by 10, we get 3, 4, 5. So scaling down by a factor of 1/10. So 0.3² + 0.4² should be 0.09 + 0.16 = 0.25, which is 0.5². Yep, 0.25. So that works too. So C is a right triangle.Now, option A: √3, √4, √5. Let's calculate those. √4 is 2, so the sides are approximately 1.732, 2, and 2.236. Let's see if the sum of the squares of the two shorter sides equals the square of the longest one.First, square each term:(√3)² = 3(√4)² = 4(√5)² = 5So, the two shorter sides squared would be 3 and 4. Their sum is 3 + 4 = 7. The hypotenuse squared is 5. But 7 ≠ 5. Wait, that doesn't add up. So that means the Pythagorean theorem doesn't hold here. Therefore, set A cannot be the lengths of a right triangle.But wait a second, maybe I mixed up the hypotenuse? Let me check the order. The problem says "sets of numbers," but it doesn't specify if they're in order. So I need to make sure which one is the longest side. The numbers given in A are √3, √4, √5. Since √5 is approximately 2.236, which is larger than √4 (which is 2) and √3 (about 1.732). So √5 is the hypotenuse. Then the legs would be √3 and √4. So as before, 3 + 4 = 7 ≠ 5. So that's definitely not a right triangle.Therefore, the answer must be option A. Let me just double-check the others quickly again. B, C, D all worked when scaled appropriately. So yeah, A is the one that doesn't satisfy the Pythagorean theorem.Final Answerboxed{A}</think>To determine which set of numbers cannot be the lengths of the sides of a right triangle, we need to check each set using the Pythagorean theorem, which states that for a right triangle, the sum of the squares of the two shorter sides must equal the square of the longest side (hypotenuse).- Option B: 3, 4, 5 - Check: (3^2 + 4^2 = 9 + 16 = 25) - Hypotenuse: (5^2 = 25) - Result: Valid right triangle.- Option D: 30, 40, 50 - Check: (30^2 + 40^2 = 900 + 1600 = 2500) - Hypotenuse: (50^2 = 2500) - Result: Valid right triangle.- Option C: 0.3, 0.4, 0.5 - Check: (0.3^2 + 0.4^2 = 0.09 + 0.16 = 0.25) - Hypotenuse: (0.5^2 = 0.25) - Result: Valid right triangle.- Option A: (sqrt{3}), (sqrt{4}), (sqrt{5}) - Check: ((sqrt{3})^2 + (sqrt{4})^2 = 3 + 4 = 7) - Hypotenuse: ((sqrt{5})^2 = 5) - Result: (7 neq 5), so not a valid right triangle.Thus, the set of numbers that cannot be the lengths of the sides of a right triangle is (boxed{A}).]

🔑:To determine which set of numbers cannot be the lengths of the sides of a right triangle, we apply the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.Option A: sqrt{3}, sqrt{4}, sqrt{5}We check if these lengths satisfy the Pythagorean theorem:[(sqrt{3})^2 + (sqrt{4})^2 stackrel{?}{=} (sqrt{5})^2][3 + 4 stackrel{?}{=} 5][7 neq 5]Since the sum of the squares of the two smaller sides does not equal the square of the largest side, these lengths cannot form a right triangle.Option B: 3, 4, 5We apply the Pythagorean theorem:[3^2 + 4^2 stackrel{?}{=} 5^2][9 + 16 = 25][25 = 25]The lengths satisfy the Pythagorean theorem, indicating they can form a right triangle.Option C: 0.3, 0.4, 0.5Checking with the Pythagorean theorem:[0.3^2 + 0.4^2 stackrel{?}{=} 0.5^2][0.09 + 0.16 = 0.25][0.25 = 0.25]These lengths also satisfy the Pythagorean theorem, so they can form a right triangle.Option D: 30, 40, 50Applying the Pythagorean theorem:[30^2 + 40^2 stackrel{?}{=} 50^2][900 + 1600 = 2500][2500 = 2500]These lengths satisfy the Pythagorean theorem, indicating they can form a right triangle.Conclusion:Only the set of numbers in Option A does not satisfy the Pythagorean theorem and thus cannot be the lengths of the sides of a right triangle. Therefore, the correct answer is boxed{A}.

🧠:[<think>Okay, so I need to find f(f(2)) for this piecewise function. Let me start by understanding the function f(x). It has two cases: when x is less than 1, it's 2x - 1, and when x is greater than or equal to 1, it's 1/x. First, I need to compute the inner function f(2). Then, whatever result I get from that, I'll plug it back into f to find the final answer. Let's break it down step by step.Starting with f(2). The input here is 2. Now, I check which case of the piecewise function applies. The function is defined as 2x - 1 when x < 1 and 1/x when x ≥ 1. Since 2 is definitely greater than or equal to 1, I should use the second case, which is 1/x. Therefore, f(2) = 1/2. That seems straightforward.Now, I need to compute f(f(2)), which is f(1/2). Wait, f(2) was 1/2, so now we plug 1/2 into the function f again. Let's see: for the input 1/2, which is 0.5, that is less than 1. So we use the first case of the function, which is 2x - 1. Calculating that: 2*(1/2) - 1. Let me compute 2*(1/2) first. That's 1. Then subtract 1, which gives 0. So f(1/2) is 0. Therefore, f(f(2)) = 0. Wait, but let me check again to make sure I didn't make a mistake. Let me go through each step once more.First, f(2): since 2 ≥ 1, use 1/x. So 1/2. Correct. Then f(1/2): 1/2 is less than 1, so use 2x - 1. 2*(1/2) is 1. 1 - 1 is 0. So yes, f(f(2)) is 0. That would correspond to option A. But let me verify once again to be absolutely sure. Maybe I misapplied the function somewhere. Let's confirm the definitions. The function f(x) is 2x -1 for x <1 and 1/x for x ≥1. First, f(2). Since 2 is not less than 1, we use the second part: 1/2. Then f(1/2). 1/2 is less than 1, so we use the first part: 2*(1/2) -1 = 1 -1 =0. So yes, 0. Hmm, the options are A:0, B:1/2, C:1, D:2. So the answer must be A. Wait, but maybe there's a trick here? Let me check if there's any alternative interpretation. For example, could there be a mistake in the order of operations or in the function definitions?No, the problem is straightforward. f is defined piecewise, with different expressions depending on the value of x. The first step is definitely f(2) =1/2. Then f(1/2) is 2*(1/2)-1=0. So yes, 0 is the answer. Alternatively, maybe the problem expects something else? For instance, if there was a different definition for f(f(x)), but the problem says f(f(2)), so it's just composition. Alternatively, could there be a mistake in the cases? Like, maybe the first case is for x ≤1? Wait, let me check the problem statement again. The function is defined as 2x -1 for x <1, and 1/x for x ≥1. So the split is at x=1. So for x=1, we use 1/x. Therefore, x=1 would be 1/1=1. But in our case, first we have x=2, which is in the second case. Then x=1/2, which is in the first case. So the calculation seems correct. Alternatively, maybe the question is written in Chinese or another language, since the original problem has Chinese parentheses? Wait, no, the problem is presented in English here, so probably not an issue. Alternatively, perhaps I misread the function definitions? Let me check again. The function f(x) is 2x-1 for x <1, and 1/x for x ≥1. Correct. Therefore, f(2)=1/2, then f(1/2)=0. Therefore, answer is A:0. Wait, but the options given are A:0, B:1/2, C:1, D:2. So 0 is an option. Therefore, the answer should be A. But I wonder if there's a common mistake here. For example, someone might compute f(2) as 1/2, then when plugging into f again, they might use the wrong case. For example, if they mistakenly thought that 1/2 is greater than or equal to 1, which it's not. So they might do 1 divided by (1/2), which is 2, leading to answer D. But that's incorrect because 1/2 is less than 1, so you have to use the first case. Alternatively, someone might miscalculate 2*(1/2) -1. Let's see: 2*(1/2) is 1. 1 -1 is 0. That's correct. So no, if someone thought 2*(1/2) is 1/1, which is 1, then subtract 1 gives 0. So that's correct. Alternatively, if someone thought that the first case is for x <=1, then for x=1, they would use 2x -1, which is 1. But in the problem statement, the first case is x <1, so x=1 is in the second case. However, in our problem, we never evaluate at x=1. We evaluated at x=2 and x=1/2. So no confusion there. Therefore, I think my reasoning is solid. The answer is A:0.Final Answerboxed{A}</think>To find ( f(f(2)) ) for the given piecewise function ( f(x) ), we proceed as follows:1. Compute ( f(2) ): - Since ( 2 geq 1 ), we use the second case of the function: ( frac{1}{x} ). - Therefore, ( f(2) = frac{1}{2} ).2. Compute ( fleft( frac{1}{2} right) ): - Since ( frac{1}{2} < 1 ), we use the first case of the function: ( 2x - 1 ). - Therefore, ( fleft( frac{1}{2} right) = 2 left( frac{1}{2} right) - 1 = 1 - 1 = 0 ).Thus, ( f(f(2)) = 0 ).The final answer is (boxed{A}).,<think>Okay, let me try to solve this problem. The function f(x) is defined piecewise, which means it has different expressions depending on the value of x. So, f(x) equals 2x - 1 when x is less than 1, and it equals 1/x when x is greater than or equal to 1. The question is asking for f(f(2)), which means we need to apply the function f twice. First, compute f(2), and then plug that result back into f.Alright, let's start with the inner function: f(2). Since 2 is the input here, I need to check which part of the piecewise function applies. The function is divided into two cases: one for x < 1 and another for x ≥ 1. The number 2 is definitely greater than 1, so we use the second case, which is 1/x. Therefore, f(2) should be 1 divided by 2, which is 1/2. Let me write that down:f(2) = 1/2.Now, we need to find f(f(2)) which is f(1/2). So now, we evaluate the function f at x = 1/2. Again, we check which case of the piecewise function applies here. The value 1/2 is 0.5, which is less than 1, so we use the first case of the function, which is 2x - 1. Plugging in 1/2 for x, we get:f(1/2) = 2*(1/2) - 1.Calculating that, 2*(1/2) is 1, and then subtract 1 gives 0. So, f(1/2) = 0.Therefore, f(f(2)) = f(1/2) = 0.Looking at the answer choices, option A is 0. So the answer should be A.But wait, let me double-check to make sure I didn't make any mistakes. Let me go through each step again.First, evaluating f(2). Since 2 ≥ 1, we use the second case: 1/x. 1 divided by 2 is indeed 1/2. That seems right.Then, take that result, which is 1/2, and plug it into f again. Now, 1/2 is less than 1, so we use the first case: 2x - 1. Substituting x with 1/2, 2*(1/2) is 1, subtract 1 gives 0. So yes, that's correct.Hmm, so the final answer is 0, which is option A. Wait, but let me check the options again. The choices are A: 0, B: 1/2, C:1, D:2. So yes, A is the correct answer. Just to be thorough, let me think if there's any chance I mixed up the cases. For example, could I have used the wrong expression for x < 1 or x ≥ 1? Let me check the original function again. The function is defined as 2x -1 for x <1 and 1/x for x ≥1. So when x is 2, which is greater than or equal to 1, we use 1/x. That gives 1/2. Then, 1/2 is less than 1, so we use 2x -1. Yep, that's 2*(1/2) -1 = 1 -1 =0. I think that's solid. No mistakes there. So the answer is A.Final Answerboxed{A}</think>To find ( f(f(2)) ), we start by evaluating the inner function ( f(2) ). Since ( 2 geq 1 ), we use the second case of the piecewise function:[f(2) = frac{1}{2}]Next, we need to evaluate ( fleft( frac{1}{2} right) ). Since ( frac{1}{2} < 1 ), we use the first case of the piecewise function:[fleft( frac{1}{2} right) = 2 left( frac{1}{2} right) - 1 = 1 - 1 = 0]Thus, ( f(f(2)) = 0 ). The correct answer is (boxed{A}).]

🔑:Since f(2) = frac{1}{2}, and fleft(frac{1}{2}right) = 2timesfrac{1}{2}-1 = 1-1 = 0,Therefore, f(f(2)) = 0.Hence, the correct choice is: boxed{A}.This problem mainly tests the calculation of function values. Using substitution based on the piecewise function expression is key to solving this problem. It is quite basic.

🧠:[<think>Alright, so I have this equation to solve: (x⁴ + x + 1)(cube root of 80 minus cube root of 0.01) equals 2 times (cube root of 5.12 plus cube root of 0.03375). Hmm, that looks a bit intimidating with all the cube roots, but maybe there's a way to simplify those terms first before dealing with the polynomial part. Let me start by focusing on the cube roots.First, let's look at the left side of the equation: (x⁴ + x + 1) multiplied by (³√80 - ³√0.01). The right side is 2 times (³√5.12 + ³√0.03375). I need to simplify these cube roots to see if they can be expressed in terms of simpler numbers or maybe related to each other.Starting with ³√80. Let me see if 80 is a multiple of a perfect cube. The cube of 4 is 64, and 80 is 64 + 16. Not sure if that helps. How about factoring 80 into prime factors: 80 = 8 * 10 = 8 * 2 * 5. So 8 is 2³, so maybe 80 = 2³ * 10. Therefore, ³√80 = ³√(2³ * 10) = 2 * ³√10. That's a start. So ³√80 is 2 * ³√10.Next, ³√0.01. Well, 0.01 is 1/100, so ³√(1/100) = 1/³√100. But 100 is 10², so ³√100 = (10²)^(1/3) = 10^(2/3). So ³√0.01 is 10^(-2/3). Alternatively, maybe express 0.01 as 10^(-2), so cube root is 10^(-2/3). Not sure if that's helpful yet. Let me note that as 10^(-2/3).So the left side becomes (x⁴ + x + 1)*(2 * ³√10 - 10^(-2/3)). Let's hold onto that for a moment.Now the right side: 2*(³√5.12 + ³√0.03375). Let's try to simplify each cube root here.First, 5.12. 5.12 is 512 divided by 100, right? 512 ÷ 100 = 5.12. And 512 is 8³, since 8*8*8=512. So ³√5.12 = ³√(512/100) = ³√512 / ³√100 = 8 / ³√100. But ³√100 is the same as 10^(2/3), so this is 8 / 10^(2/3) = 8 * 10^(-2/3). Hmm, interesting. Let me write that down.Next, ³√0.03375. Let's see. 0.03375. Hmm, that's 3375 divided by 100,000. Because 3375 * 0.00001 = 0.03375. Wait, 3375 is 15³, since 15*15=225, 225*15=3375. So ³√3375 = 15. Therefore, ³√(0.03375) = ³√(3375 / 100,000) = 15 / ³√100,000. But 100,000 is 10^5, so ³√10^5 = 10^(5/3). Therefore, ³√0.03375 = 15 / 10^(5/3) = 15 * 10^(-5/3). Alternatively, 15 divided by 10^(1 + 2/3) = 15 / (10 * 10^(2/3)) ) = (15/10) / 10^(2/3) = 1.5 / 10^(2/3) = 1.5 * 10^(-2/3).So now, let's rewrite the right side: 2*(8 * 10^(-2/3) + 1.5 * 10^(-2/3)). Since both terms have 10^(-2/3), we can factor that out. So 2*( (8 + 1.5) * 10^(-2/3) ) = 2*(9.5 * 10^(-2/3)) = 19 * 10^(-2/3).So the right side simplifies to 19 * 10^(-2/3). Let me check that calculation again. 8 + 1.5 is indeed 9.5, multiplied by 2 is 19. So yes.Now looking back at the left side: (x⁴ + x + 1)*(2 * ³√10 - 10^(-2/3)). Let's see if we can factor out 10^(-2/3) here as well. Let's write both terms in the parentheses with exponent -2/3.First, 2 * ³√10. ³√10 is 10^(1/3), so 2*10^(1/3). Then 10^(-2/3) is already as is. So we have 2*10^(1/3) - 10^(-2/3). Let's factor out 10^(-2/3). Then we get 10^(-2/3)*(2*10^(1/3 + 2/3) - 1). Wait, exponent rules: when factoring out 10^(-2/3), the remaining exponent for the first term is 1/3 - (-2/3) = 1/3 + 2/3 = 1. So 2*10^(1) - 1. Therefore, the left side becomes (x⁴ + x + 1)*10^(-2/3)*(2*10 - 1). 2*10 is 20, minus 1 is 19. So the left side simplifies to (x⁴ + x + 1)*10^(-2/3)*19.So putting it all together, the equation is:(x⁴ + x + 1)*19*10^(-2/3) = 19*10^(-2/3)Wait, because the right side was 19*10^(-2/3). So if I divide both sides by 19*10^(-2/3), assuming it's non-zero (which it is, since 10^(-2/3) is positive), then we get:x⁴ + x + 1 = 1So then, x⁴ + x + 1 = 1 implies x⁴ + x = 0.Factoring that, x(x³ + 1) = 0.So the solutions are x = 0 or x³ + 1 = 0.x³ + 1 = 0 implies x³ = -1, so x = -1.Therefore, the solutions are x = 0 or x = -1.But wait, let me double-check that. The original equation had cube roots, which are defined for all real numbers, so we don't have to worry about taking roots of negative numbers here. The polynomial x⁴ + x + 1 evaluated at x=0 is 0 + 0 + 1 = 1, and at x=-1 is (-1)^4 + (-1) + 1 = 1 -1 +1 = 1. So both x=0 and x=-1 make the left side (x⁴ +x +1) equal to 1, which when multiplied by that factor gives 19*10^(-2/3), same as the right side. So they are valid solutions.But wait, let's check if there are any other solutions. When we divided both sides by 19*10^(-2/3), we assumed that 19*10^(-2/3) ≠ 0, which is true. So we didn't lose any solutions there. So the only solutions are x=0 and x=-1.But hold on, the polynomial x⁴ + x + 1. If we set x⁴ + x +1 =1, then x⁴ +x =0, so x(x³ +1)=0. So x=0 or x³ = -1, so x=-1. Are there any complex solutions? The problem doesn't specify to find only real solutions, but in the context of cube roots, unless specified, sometimes they refer to real roots. However, the original equation didn't specify, but since we have real coefficients and real cube roots, maybe they expect real solutions. So x=0 and x=-1. Let me verify by plugging them into the original equation.First, x=0:Left side: (0⁴ +0 +1)(³√80 - ³√0.01) = (1)(³√80 - ³√0.01)Right side: 2(³√5.12 + ³√0.03375)Earlier, we simplified both sides to 19*10^(-2/3), so both sides are equal. So x=0 works.Now x=-1:Left side: ((-1)^4 + (-1) +1)(³√80 - ³√0.01) = (1 -1 +1)(same cube roots) = 1*(³√80 - ³√0.01) = same as before, which equals the right side. So x=-1 also works.Therefore, the solutions are x=0 and x=-1. But wait, let me check if there's a mistake in simplification. Because when we had x⁴ + x +1 =1 leading to x⁴ +x =0, so x=0 or x³=-1. So x=0 and x=-1 are the real solutions. The other solutions of x³ = -1 are complex, but since we're dealing with real numbers here (cube roots of real numbers are real), the equation is in the real number system, so the only real solutions are x=0 and x=-1. If the problem allows complex solutions, then there would be more, but likely they want real solutions.Therefore, the solutions are x=0 and x=-1.Final AnswerThe solutions are boxed{0} and boxed{-1}.</think>To solve the equation (left(x^{4}+x+1right)(sqrt[3]{80}-sqrt[3]{0.01})=2(sqrt[3]{5.12}+sqrt[3]{0.03375})), we start by simplifying the cube roots on both sides.First, we simplify the left side:- (sqrt[3]{80}) can be written as (2 sqrt[3]{10}) because (80 = 2^3 cdot 10).- (sqrt[3]{0.01}) is (10^{-2/3}) because (0.01 = 10^{-2}).Thus, the left side becomes:[(x^4 + x + 1) left(2 sqrt[3]{10} - 10^{-2/3}right)]Next, we simplify the right side:- (sqrt[3]{5.12}) is (frac{8}{10^{2/3}}) because (5.12 = frac{512}{100}) and (sqrt[3]{512} = 8).- (sqrt[3]{0.03375}) is (frac{15}{10^{5/3}} = 1.5 cdot 10^{-2/3}) because (0.03375 = frac{3375}{100000}) and (sqrt[3]{3375} = 15).Thus, the right side becomes:[2 left(8 cdot 10^{-2/3} + 1.5 cdot 10^{-2/3}right) = 2 left(9.5 cdot 10^{-2/3}right) = 19 cdot 10^{-2/3}]Now, we factor out (10^{-2/3}) from the left side:[(x^4 + x + 1) left(10^{-2/3} (2 cdot 10 - 1)right) = (x^4 + x + 1) cdot 10^{-2/3} cdot 19]Equating both sides, we get:[(x^4 + x + 1) cdot 19 cdot 10^{-2/3} = 19 cdot 10^{-2/3}]Dividing both sides by (19 cdot 10^{-2/3}), we obtain:[x^4 + x + 1 = 1]This simplifies to:[x^4 + x = 0]Factoring out (x), we get:[x(x^3 + 1) = 0]This gives the solutions:[x = 0 quad text{or} quad x^3 = -1 implies x = -1]Thus, the solutions to the equation are (boxed{0}) and (boxed{-1}).,<think>Okay, let me try to solve this equation step by step. The equation given is:[left(x^{4} + x + 1right)left(sqrt[3]{80} - sqrt[3]{0.01}right) = 2left(sqrt[3]{5.12} + sqrt[3]{0.03375}right)]Hmm, so first, I need to simplify both sides of the equation, probably starting with the cube roots. Let me see if I can express those cube roots in terms of simpler numbers or maybe relate them to each other. Starting with the left side: (sqrt[3]{80} - sqrt[3]{0.01}). Let me see if these can be rewritten. 80 is 16 times 5, which is 2^4 * 5, but I'm not sure if that helps. 0.01 is 1/100, so the cube root of 1/100. Maybe I can write them as fractions. Wait, 80 is also 8 * 10, so 8 is 2^3. So, (sqrt[3]{80} = sqrt[3]{8 times 10} = sqrt[3]{8} times sqrt[3]{10} = 2 times sqrt[3]{10}). Similarly, (sqrt[3]{0.01} = sqrt[3]{frac{1}{100}} = frac{1}{sqrt[3]{100}}). But 100 is 10^2, so (sqrt[3]{100} = (sqrt[3]{10})^2). Hmm, not sure if that helps yet. Maybe let's compute approximate values to see if they can be simplified.Alternatively, maybe the terms on both sides can be expressed in terms of some common variables. Let me check the right side as well: (2(sqrt[3]{5.12} + sqrt[3]{0.03375})).5.12 and 0.03375... Let me see. 5.12 is 512 divided by 100, right? 512 is 8^3, so 512/100 = 8^3 / 10^2. Therefore, (sqrt[3]{5.12} = sqrt[3]{512/100} = sqrt[3]{8^3 / 10^2} = 8 / sqrt[3]{10^2} = 8 / (10^{2/3})). Similarly, 0.03375 is 33.75/1000. Let me check 33.75. Hmm, 33.75 is 27 + 6.75 = 27 + 6.75. Wait, 27 is 3^3, and 6.75 is 27/4, so 33.75 is 27 + 27/4 = 27*(1 + 1/4) = 27*(5/4). Therefore, 33.75 = (27*5)/4. So 0.03375 is (27*5)/(4*1000) = 135/4000 = 27/800. Let me confirm that:27/800: 27 divided by 800. 27/8 = 3.375, so 27/800 = 0.03375. Correct. So (sqrt[3]{0.03375} = sqrt[3]{27/800}). 27 is 3^3, and 800 is 8*100, which is 8*10^2. Therefore, (sqrt[3]{27/800} = sqrt[3]{27}/sqrt[3]{800} = 3/sqrt[3]{8 times 100} = 3/(2 times sqrt[3]{100})). So that's 3/(2 * 10^{2/3}).Wait, so putting that together, the right side is:2*(8 / 10^{2/3} + 3/(2 * 10^{2/3})) = 2*(8 / 10^{2/3} + 3/(2 * 10^{2/3})).Let me compute this:First term inside the parentheses: 8 / 10^{2/3}Second term: 3/(2 * 10^{2/3})Adding them: 8 / 10^{2/3} + 3/(2 * 10^{2/3}) = (16/2 + 3/2) / 10^{2/3} = (19/2) / 10^{2/3} = 19/(2 * 10^{2/3})Multiply by 2: 2 * (19)/(2 * 10^{2/3}) = 19 / 10^{2/3}So the right side simplifies to 19 / 10^{2/3}Now, let's look back at the left side. The left side is (x^4 + x + 1)(sqrt[3]{80} - sqrt[3]{0.01})Earlier, I tried to express sqrt[3]{80} as 2 * sqrt[3]{10} and sqrt[3]{0.01} as 1 / sqrt[3]{100} = 1 / (10^{2/3})Therefore, sqrt[3]{80} - sqrt[3]{0.01} = 2 * 10^{1/3} - 10^{-2/3}Hmm, let me write that as 2 * 10^{1/3} - 10^{-2/3}Alternatively, factor out 10^{-2/3}:10^{-2/3} (2 * 10^{(1/3 + 2/3)} - 1) = 10^{-2/3} (2 * 10^{1} - 1) = 10^{-2/3} (20 - 1) = 19 * 10^{-2/3}So that simplifies the left side's coefficient to 19 * 10^{-2/3}Therefore, the left side is (x^4 + x + 1) * 19 * 10^{-2/3}So, putting it all together, the equation becomes:(x^4 + x + 1) * 19 * 10^{-2/3} = 19 / 10^{2/3}Divide both sides by 19 * 10^{-2/3}, assuming it's non-zero (which it is, since 19 is non-zero and 10^{-2/3} is positive):(x^4 + x + 1) = (19 / 10^{2/3}) / (19 * 10^{-2/3}) ) = (19 / 10^{2/3}) * (1 / (19 * 10^{-2/3})) ) = (1 / 10^{2/3}) / (10^{-2/3}) ) = 1 / 10^{2/3} * 10^{2/3} = 1Therefore, x^4 + x + 1 = 1Subtract 1 from both sides:x^4 + x = 0Factor out x:x(x^3 + 1) = 0So, either x = 0 or x^3 + 1 = 0If x^3 + 1 = 0, then x^3 = -1 => x = -1Therefore, the solutions are x = 0 or x = -1But wait, we need to check if these solutions actually satisfy the original equation, because sometimes when we manipulate equations, especially involving cube roots or other operations, extraneous solutions might pop up. Let's check both solutions in the original equation.First, check x = 0:Left side: (0^4 + 0 + 1)(sqrt[3]{80} - sqrt[3]{0.01}) = (1)(sqrt[3]{80} - sqrt[3]{0.01}) = sqrt[3]{80} - sqrt[3]{0.01}Right side: 2(sqrt[3]{5.12} + sqrt[3]{0.03375})We need to check if sqrt[3]{80} - sqrt[3]{0.01} equals 2(sqrt[3]{5.12} + sqrt[3]{0.03375})But earlier, we simplified both sides to 19 / 10^{2/3}, so they should be equal. Let me confirm with actual numbers:Compute left side: sqrt[3]{80} ≈ cube root of 80 ≈ 4.308, sqrt[3]{0.01} ≈ 0.215. So 4.308 - 0.215 ≈ 4.093Right side: 2*(sqrt[3]{5.12} + sqrt[3]{0.03375}) ≈ 2*(1.724 + 0.324) ≈ 2*(2.048) ≈ 4.096So approximately 4.093 ≈ 4.096. Close enough, considering rounding errors. So x = 0 is a valid solution.Now check x = -1:Left side: ((-1)^4 + (-1) + 1)(sqrt[3]{80} - sqrt[3]{0.01}) = (1 -1 + 1)(...) = (1)(sqrt[3]{80} - sqrt[3]{0.01}) ≈ same as before, 4.093Right side: same as before, ≈4.096Wait, so even for x = -1, the left side is the same as when x = 0? Because x^4 + x + 1 when x = -1 is 1 + (-1) + 1 = 1. So indeed, same as x = 0, so x = -1 also gives the same value. Therefore, both x = 0 and x = -1 satisfy the equation. Therefore, both are solutions.Wait, but in the algebra earlier, we had x^4 + x + 1 = 1, leading to x^4 + x = 0, which factors as x(x^3 + 1) = 0. So the solutions are x = 0 and x^3 = -1 => x = -1. So both solutions are valid.But just to be thorough, let's compute x = -1 in the original equation:Left side: ((-1)^4 + (-1) + 1)(sqrt[3]{80} - sqrt[3]{0.01}) = (1 -1 + 1)(4.308 - 0.215) = 1 * 4.093 ≈ 4.093Right side: 2*(sqrt[3]{5.12} + sqrt[3]{0.03375}) ≈ 4.096 as before. So the slight difference is due to approximate calculations. If we do exact calculations, they should be equal. Since we derived algebraically that both sides reduce to the same expression, both x = 0 and x = -1 are indeed solutions.Therefore, the solutions are x = 0 and x = -1.Final AnswerThe solutions are (boxed{0}) and (boxed{-1}).</think>To solve the equation (left(x^{4} + x + 1right)left(sqrt[3]{80} - sqrt[3]{0.01}right) = 2left(sqrt[3]{5.12} + sqrt[3]{0.03375}right)), we start by simplifying both sides.First, we simplify the cube roots on both sides. For the left side:[sqrt[3]{80} - sqrt[3]{0.01} = 2 sqrt[3]{10} - frac{1}{sqrt[3]{100}} = 2 cdot 10^{1/3} - 10^{-2/3}]Factoring out (10^{-2/3}):[10^{-2/3} (2 cdot 10 - 1) = 10^{-2/3} cdot 19]Thus, the left side becomes:[(x^4 + x + 1) cdot 19 cdot 10^{-2/3}]For the right side, we simplify:[2left(sqrt[3]{5.12} + sqrt[3]{0.03375}right) = 2 left( frac{8}{10^{2/3}} + frac{3}{2 cdot 10^{2/3}} right) = 2 left( frac{19}{2 cdot 10^{2/3}} right) = frac{19}{10^{2/3}}]Equating both sides:[(x^4 + x + 1) cdot 19 cdot 10^{-2/3} = frac{19}{10^{2/3}}]Dividing both sides by (19 cdot 10^{-2/3}):[x^4 + x + 1 = 1]Subtracting 1 from both sides:[x^4 + x = 0]Factoring out (x):[x(x^3 + 1) = 0]This gives the solutions (x = 0) or (x^3 = -1) (which implies (x = -1)).Both solutions are verified to satisfy the original equation. Therefore, the solutions are:[boxed{0} text{ and } boxed{-1}]]

🔑:Answer: x_{1}=0, x_{2}=-1. Solution. See problem 10.1.

🧠:[<think>Okay, so I need to find all integer pairs (a, b) such that a(a - b) = b. Hmm, let's start by understanding what this equation means. The left side is a multiplied by (a minus b), and the right side is just b. Since a and b are integers, maybe I can manipulate this equation to express it in terms that are easier to handle, like factoring or something.Let me rewrite the equation:a(a - b) = bExpanding the left side gives me:a² - ab = bHmm, maybe I can bring all terms to one side to set the equation to zero. Let's subtract b from both sides:a² - ab - b = 0Now, looking at this, maybe I can factor out a common term. The first two terms have an 'a', and the last term is -b. Let me see:a² - ab - b = a(a - b) - b = 0But that just takes me back to the original equation. Maybe another approach. Let's factor out 'b' from the last two terms:a² - b(a + 1) = 0Hmm, so a² = b(a + 1). That seems useful. So from this, I can solve for b:b = a² / (a + 1)But since b has to be an integer, this fraction must simplify to an integer. So for each integer a (except a = -1, because denominator is zero), we need a² divisible by (a + 1). So, (a + 1) divides a². Let's denote d = a + 1. Then, a = d - 1. Substituting back into the equation:b = (d - 1)² / dExpanding the numerator:(d² - 2d + 1) / d = d - 2 + 1/dSo for this to be an integer, 1/d must be an integer. Since d is an integer (because a is integer, so d = a + 1 is also integer), 1/d is integer only if d is a divisor of 1. So the divisors of 1 are 1 and -1. Therefore, d can be 1 or -1.So let's consider the possible values of d:Case 1: d = 1Then, a = d - 1 = 1 - 1 = 0Then, b = (0)^2 / (0 + 1) = 0 / 1 = 0So pair (0, 0) is a solution.Case 2: d = -1Then, a = d - 1 = -1 - 1 = -2Then, b = (-2)^2 / (-2 + 1) = 4 / (-1) = -4So pair (-2, -4) is another solution.Wait, but is that all? Because we considered d as divisors of 1, but maybe I missed something here. Let me check again.We said that d must divide 1, so d is ±1. But perhaps there's another way to approach this. Let me verify with another method.Original equation: a(a - b) = bLet me rearrange to express b in terms of a:From a(a - b) = bMultiply out:a² - ab = bBring all terms to left:a² - ab - b = 0Factor b:a² - b(a + 1) = 0So, a² = b(a + 1)Therefore, b = a² / (a + 1)But for b to be integer, (a + 1) must divide a². So (a + 1) divides a². Let's think about divisors. For integers, if m divides n, then n = k*m for some integer k.So, (a + 1) divides a², so there exists integer k such that a² = k*(a + 1). Then, we can write this as quadratic equation in a:a² - k*a - k = 0But maybe this is complicating. Alternatively, since (a + 1) divides a², we can use the division algorithm or gcd.Note that gcd(a + 1, a). Since a and a + 1 are consecutive integers, their gcd is 1. Therefore, since a + 1 divides a² and gcd(a + 1, a) = 1, then a + 1 divides 1. Wait, that seems like a standard number theory approach.Wait, let's formalize it. If d = a + 1, then we have d divides a². But since d and a are consecutive integers, gcd(d, a) = gcd(a + 1, a) = gcd(a, 1) = 1. Therefore, since d divides a² and gcd(d, a) = 1, then d divides 1. Hence, d is ±1. So d = 1 or d = -1. Which leads to a = 0 or a = -2, as before. Therefore, the only possible a are 0 and -2, leading to b = 0 and b = -4, respectively.But let me test these solutions to make sure.First pair (0, 0):Left side: 0*(0 - 0) = 0*0 = 0. Right side: 0. So that's correct.Second pair (-2, -4):Left side: -2*(-2 - (-4)) = -2*(-2 + 4) = -2*(2) = -4. Right side: -4. Correct.But wait, is there any other possible solution?Suppose we try other values of a and see if b is integer.Let's try a = 1:Then b = 1² / (1 + 1) = 1/2. Not integer. So no solution.a = 2:b = 4 / 3. Not integer.a = -1:Original equation: a(a - b) = b. If a = -1, then (-1)(-1 - b) = b. So (-1)(-1 - b) = (1 + b) = b. So 1 + b = b implies 1 = 0, which is impossible. So a = -1 is invalid.a = -3:b = (-3)^2 / (-3 +1 ) = 9 / (-2) = -9/2. Not integer.a = 3:9 / 4. Not integer.a = 4:16 / 5. Not integer.a = -4:16 / (-3) ≈ -5.333. Not integer.So indeed, only a = 0 and a = -2 give integer b. Therefore, the solutions are (0,0) and (-2, -4). Wait, but let me check if there are other divisors of a² that are a +1. Wait, perhaps if a +1 divides a², even if a +1 is not ±1? But according to the previous logic, since a +1 and a are coprime, a +1 must divide 1, hence only ±1. So unless that logic is wrong.Alternatively, maybe if a +1 divides a², even if they are not coprime? Wait, but the logic is that since a and a +1 are coprime, then a +1 divides a² implies a +1 divides 1. Let me check again.Suppose that a +1 divides a². Since gcd(a, a +1) =1, then gcd(a +1, a²) = gcd(a +1, a^2). But since gcd(a +1, a) =1, then gcd(a +1, a²) =1 as well. Therefore, a +1 divides 1. Hence, a +1 must be ±1. Therefore, a +1 =1 => a=0, or a +1 = -1 => a=-2.Hence, the only possible solutions. Therefore, only two pairs: (0,0) and (-2, -4). So that's the answer.But wait, let me check if (0,0) is acceptable. If a=0 and b=0, then the equation is 0*(0 -0)=0, which is 0=0. Correct. Similarly, a=-2 and b=-4: -2*(-2 - (-4)) = -2*(-2 +4)= -2*(2) = -4, which equals b=-4. Correct.Is there any other way this equation can hold? Let's try substituting b from the equation into itself. Starting again with a(a - b) = b. Let me solve for b:a² - ab = ba² = ab + ba² = b(a +1)So, as before, b = a²/(a +1). So we can also think of this as for each integer a ≠ -1, compute b and check if it's integer. So for a ≠ -1, b must be integer. So perhaps there's a pattern here.Alternatively, maybe we can set k = a +1, then a = k -1, and then express b in terms of k. Let's try that substitution:a = k -1Then, b = a²/(a +1) = (k -1)^2 / kExpanding numerator:(k² - 2k +1)/k = k - 2 + 1/kSo for b to be integer, 1/k must be integer. Therefore, k must divide 1. So k is ±1. Therefore, k=1 gives a=0, b=0. k=-1 gives a=-2, b=-4. So same result.Thus, confirming that only these two solutions exist. Therefore, the integer pairs are (0,0) and (-2,-4). So I think that's all.Final AnswerThe integer pairs are boxed{(0, 0)} and boxed{(-2, -4)}.</think>To determine all pairs ((a, b)) of integers such that (a(a - b) = b), we start by rewriting the equation:[a(a - b) = b]Expanding and rearranging terms, we get:[a^2 - ab - b = 0]Factoring out (b) from the last two terms, we have:[a^2 = b(a + 1)]Solving for (b), we get:[b = frac{a^2}{a + 1}]For (b) to be an integer, (frac{a^2}{a + 1}) must be an integer. This implies that (a + 1) must divide (a^2). Let (d = a + 1). Then (a = d - 1), and substituting back, we get:[b = frac{(d - 1)^2}{d}]Expanding the numerator, we have:[b = frac{d^2 - 2d + 1}{d} = d - 2 + frac{1}{d}]For (b) to be an integer, (frac{1}{d}) must be an integer. This is only possible if (d) is a divisor of 1, which means (d = 1) or (d = -1).- If (d = 1), then (a = 1 - 1 = 0) and (b = frac{0^2}{0 + 1} = 0).- If (d = -1), then (a = -1 - 1 = -2) and (b = frac{(-2)^2}{-2 + 1} = frac{4}{-1} = -4).Thus, the integer pairs ((a, b)) that satisfy the equation are ((0, 0)) and ((-2, -4)).[boxed{(0, 0)} quad text{and} quad boxed{(-2, -4)}],<think>Okay, so I need to find all integer pairs (a, b) such that a(a - b) = b. Hmm, let me start by understanding the equation better. Let's write it out again:a(a - b) = bMaybe I can expand the left side first. That gives me:a² - ab = bHmm, so if I bring all terms to one side, it becomes:a² - ab - b = 0Hmm, maybe I can factor something out here. Let's see... The first two terms have an 'a' in them, but the last term is just -b. Let me check:a² - ab - b = a(a - b) - b = 0But that's how it started. Maybe there's another way to factor. Let's see... Alternatively, maybe I can rearrange terms to isolate b?Starting from the original equation:a² - ab = bLet's move the b to the left:a² - ab - b = 0Now, factor out a b from the last two terms:a² - b(a + 1) = 0So that gives:a² = b(a + 1)So then, solving for b:b = a² / (a + 1)But since a and b are integers, this fraction must simplify to an integer. Therefore, (a + 1) must divide a². So (a + 1) | a². So we need to find all integers a such that (a + 1) divides a². Then b would be the result of that division.So the key here is to find all integers a where a + 1 divides a², and then compute b accordingly.So first, let's note that a + 1 divides a². So for each integer a (except a = -1, which would make the denominator zero), we need a + 1 divides a². Then b would be a²/(a + 1). But we need to find all such integer pairs.Alternatively, maybe we can set k = a + 1, so that a = k - 1. Then substitute into the equation for b:b = (k - 1)² / kSo expanding (k - 1)²: k² - 2k + 1. Therefore,b = (k² - 2k + 1)/k = k - 2 + 1/kBut since b has to be an integer, 1/k must be an integer. So 1/k is integer only if k divides 1. Therefore, k must be a divisor of 1. The divisors of 1 are 1 and -1. So k = 1 or k = -1.Therefore, k ∈ {1, -1}So then, if k = 1:a = k - 1 = 1 - 1 = 0Then b = (0)² / (0 + 1) = 0 / 1 = 0. So the pair (0, 0).If k = -1:a = k - 1 = -1 - 1 = -2Then b = (-2)² / (-2 + 1) = 4 / (-1) = -4. So the pair (-2, -4).Wait, but are these the only solutions? Let me check.But hold on, by substituting a = k - 1, we might have restricted ourselves? Wait, but since k = a + 1, and we require that k divides 1. So k must be ±1. Therefore, a = 0 or -2. So those are the only possible a's. Then, those are the only pairs? Let's test.First, check (0, 0):Left side: 0*(0 - 0) = 0*0 = 0. Right side: 0. So yes, that works.Second, check (-2, -4):Left side: -2*(-2 - (-4)) = -2*(-2 + 4) = -2*(2) = -4. Right side: -4. So that works too.But wait, are there any other solutions? Let's test some other values of a.Suppose a = 1. Then b = 1² / (1 + 1) = 1/2. Not integer. So (1, 1/2) is invalid.a = 2: b = 4 / 3. Not integer.a = -1: Original equation becomes -1*(-1 - b) = b. Let's compute: -1*(-1 - b) = (1 + b) = b. So 1 + b = b → 1 = 0. Contradiction. So no solution here.a = -3: Then b = (-3)^2 / (-3 + 1) = 9 / (-2) = -4.5. Not integer.a = 3: 9 / 4. Not integer.a = 4: 16 / 5. Not integer.a = -4: 16 / (-3) ≈ -5.333. Not integer.So seems like only a = 0 and a = -2 give integer b. But let me think again. Maybe there is another way to approach this.Alternatively, starting from a(a - b) = b. Let's rearrange terms to get:a² - ab - b = 0a² - b(a + 1) = 0So, as before, a² = b(a + 1). So if a + 1 divides a², then b must be a²/(a + 1). So another way to write this is:b = a² / (a + 1) = [a² - 1 + 1] / (a + 1) = [ (a - 1)(a + 1) + 1 ] / (a + 1) = a - 1 + 1/(a + 1)So this gives b = a - 1 + 1/(a + 1)Since b must be an integer, 1/(a + 1) must be integer. Therefore, 1/(a + 1) is integer only if a + 1 divides 1. Therefore, a + 1 is ±1. Hence, a + 1 = 1 ⇒ a = 0; a + 1 = -1 ⇒ a = -2. Which brings us back to the same solutions.Therefore, only possible when a = 0 or a = -2, leading to b = 0 or b = -4 respectively. Hence, the integer pairs are (0, 0) and (-2, -4). Therefore, these are the only solutions.Wait, but let me check if there's a case when a + 1 = 0, but a + 1 is zero. Then a = -1, but then the original equation becomes:a(a - b) = b-1*(-1 - b) = b ⇒ (1 + b) = b ⇒ 1 = 0, which is impossible, so a = -1 is invalid.Hence, the only possible solutions are (0, 0) and (-2, -4). But let me check if there are other possible divisors. Wait, we assumed that a + 1 divides 1, but maybe there is another way to write the equation where a + 1 divides something else.Wait, in the equation a² = b(a + 1), a + 1 divides a². So, in number theory, if d divides n², then d divides the square of n. So, the divisors of n² are all the divisors of n squared. But here, d is a + 1, and n is a. So, a + 1 divides a². So, perhaps, we can use the property that gcd(a, a + 1) = 1, since consecutive integers are coprime. Therefore, since a and a + 1 are coprime, then a + 1 must divide 1. Wait, that might be the case.Wait, if a and a + 1 are coprime, then any common divisor must divide their difference, which is 1. So, since gcd(a, a + 1) = 1, then a + 1 divides a². But since a and a + 1 are coprime, a + 1 must divide 1. Because a + 1 divides a² and gcd(a + 1, a) = 1, so a + 1 divides 1. Therefore, a + 1 must be ±1. Therefore, the conclusion is that a + 1 = ±1, leading to a = 0 or a = -2. Therefore, confirming our previous result.Therefore, the only integer solutions are (0, 0) and (-2, -4). Let me just verify again with these two pairs.For (0, 0):Left side: 0*(0 - 0) = 0*0 = 0. Right side: 0. So 0 = 0. Correct.For (-2, -4):Left side: -2*(-2 - (-4)) = -2*(-2 + 4) = -2*(2) = -4. Right side: -4. So -4 = -4. Correct.Therefore, these are indeed solutions. Are there any other possible solutions? Let's suppose that a + 1 divides a². But since a + 1 and a are coprime, so as per Bezout's identity, since they are coprime, a + 1 must divide 1. Therefore, only divisors of 1 are ±1, so a + 1 must be ±1. Therefore, only two possible values for a.Alternatively, suppose we didn't use that coprime argument. Let's consider that a + 1 divides a², so a + 1 divides a². Let’s write a² = k(a + 1) where k is an integer. Then, a² - k a - k = 0. This is a quadratic in a. For integer a, the discriminant must be a perfect square.The discriminant D = k² + 4k. So D must be a perfect square.So, k² + 4k = m² for some integer m.So, k² + 4k - m² = 0.This can be rewritten as (k + 2)^2 - m² = 4. So (k + 2 - m)(k + 2 + m) = 4.Therefore, we have two integers (k + 2 - m) and (k + 2 + m) such that their product is 4. Now, we can list all pairs of integers (s, t) such that s * t = 4, and then solve for k and m.The integer pairs (s, t) are:(1, 4), (2, 2), (-1, -4), (-2, -2), (4, 1), (-4, -1)But since s = k + 2 - m and t = k + 2 + m, then s and t must satisfy s + t = 2(k + 2). Therefore, s and t must have the same parity because their sum is even. Let's check each pair:1. (1, 4):s + t = 5. But 2(k + 2) = 5 → k + 2 = 2.5. Not integer. Discard.2. (2, 2):s + t = 4. Then 2(k + 2) = 4 → k + 2 = 2 → k = 0. Then s = 2 = k + 2 - m → 2 = 0 + 2 - m → 2 = 2 - m → m = 0. So m = 0, k = 0. Then original quadratic: a² - 0*a - 0 = a² = 0 → a = 0. So (a, b) = (0, 0). Which we have.3. (-1, -4):s + t = -5. 2(k + 2) = -5 → k + 2 = -2.5. Not integer. Discard.4. (-2, -2):s + t = -4. Then 2(k + 2) = -4 → k + 2 = -2 → k = -4. Then s = -2 = k + 2 - m → -2 = -4 + 2 - m → -2 = -2 - m → m = 0. Then original quadratic: a² - (-4)a - (-4) = a² + 4a + 4 = 0 → (a + 2)^2 = 0 → a = -2. Therefore, (a, b) = (-2, (-2)^2 / (-2 + 1)) = (-2, 4 / (-1)) = (-2, -4). Which is the other solution.5. (4, 1):Same as (1, 4). s + t = 5. Not valid.6. (-4, -1):Same as (-1, -4). s + t = -5. Not valid.So only the pairs (2, 2) and (-2, -2) give valid solutions. Therefore, only k = 0 and k = -4 lead to valid integer a's, which are a = 0 and a = -2, leading to b = 0 and b = -4. So that's another method leading to the same conclusion.Alternatively, maybe I can approach this by parametrizing the equation. Let me see.Original equation: a(a - b) = b.Let me express b in terms of a. So:b = a(a - b)But that's the original equation. Let's rearrange:b + ab = a²So:b(1 + a) = a²Therefore, b = a² / (a + 1). Which is the same as before.So, we need a² / (a + 1) to be an integer. So (a + 1) divides a².Since gcd(a, a + 1) = 1, then a + 1 must divide 1. Therefore, a + 1 = ±1. So a = 0 or a = -2.Therefore, only two solutions.Alternatively, let's suppose that (a + 1) divides a². Let d = a + 1. Then d divides a². But since d = a + 1, we can write a = d - 1. So substituting back:d divides (d - 1)^2.Therefore, d divides (d - 1)^2.Compute (d - 1)^2 = d² - 2d + 1. Therefore,d divides d² - 2d + 1. Therefore, d divides 1 (since d divides d² - 2d, and d divides the whole expression, so d divides 1). Hence, d must be ±1. Therefore, d = ±1, so a + 1 = ±1 ⇒ a = 0 or a = -2. Hence, same result.Therefore, this confirms that the only solutions are when a = 0 or a = -2, leading to b = 0 or b = -4.Wait, but let me check with a different approach. Suppose I set variables differently. Let's consider the equation as a quadratic in b:a(a - b) = b ⇒ a² - ab = b ⇒ a² = ab + b ⇒ a² = b(a + 1)So, same as before.Alternatively, considering this as a quadratic equation in b:ab + b - a² = 0 ⇒ b(a + 1) = a² ⇒ same as before.Alternatively, treating the original equation as a quadratic in a:a² - ab - b = 0Quadratic equation in a:a² - b a - b = 0Discriminant D = b² + 4bSo for a to be integer, discriminant must be a perfect square. So D = b² + 4b = k² for some integer k.So, similar to earlier, we can write this as:k² = b² + 4b ⇒ k² = (b + 2)^2 - 4So, (b + 2)^2 - k² = 4 ⇒ (b + 2 - k)(b + 2 + k) = 4Again, factor pairs of 4. Let me list all pairs (s, t) such that s * t = 4, with s ≤ t:(1, 4), (2, 2), (-1, -4), (-2, -2), (4, 1), (-4, -1)But since s = b + 2 - k and t = b + 2 + k, we have s + t = 2(b + 2) and t - s = 2k. So s and t must have the same parity because their sum and difference are both even. Let's check each pair:1. (1, 4):s + t = 5 ⇒ 2(b + 2) = 5 ⇒ b + 2 = 2.5 ⇒ b = 0.5. Not integer. Discard.2. (2, 2):s + t = 4 ⇒ 2(b + 2) = 4 ⇒ b + 2 = 2 ⇒ b = 0. Then, t - s = 0 = 2k ⇒ k = 0. Then from quadratic equation:a² - 0*a - 0 = a² = 0 ⇒ a = 0. So solution (0, 0).3. (-1, -4):s + t = -5 ⇒ 2(b + 2) = -5 ⇒ b + 2 = -2.5 ⇒ b = -4.5. Not integer. Discard.4. (-2, -2):s + t = -4 ⇒ 2(b + 2) = -4 ⇒ b + 2 = -2 ⇒ b = -4. Then, t - s = 0 = 2k ⇒ k = 0. Then quadratic equation:a² - (-4)a - (-4) = a² +4a +4 =0 ⇒ (a + 2)^2 = 0 ⇒ a = -2. So solution (-2, -4).5. (4, 1):s + t = 5 ⇒ same as case 1. Disregard.6. (-4, -1):s + t = -5 ⇒ same as case 3. Disregard.Thus, only solutions are when b = 0 and b = -4, leading to a = 0 and a = -2. So again, same pairs.Therefore, after multiple approaches, the only integer solutions are (0, 0) and (-2, -4).But just to be thorough, let me check other possible factor pairs. Wait, in the previous steps, when we set (k + 2 - m)(k + 2 + m) = 4, we considered all integer pairs. What if there are other factorizations, like (s, t) where s and t are non-integer? But since s and t must be integers (as k and m are integers), so no, only integer pairs.Therefore, only the two solutions.Alternatively, let's check for a few more a values manually.a = 5: b = 25 /6 ≈4.166, not integer.a = -6: b = 36 / (-5) = -7.2, not integer.a = 10: b = 100 /11 ≈9.09, not integer.a = -3: b = 9 / (-2) = -4.5, not integer.a = 1: b = 1/2, nope.a = 2: 4/3, nope.So, yeah, no other integer pairs found.Thus, the conclusion is that the only integer solutions are (0, 0) and (-2, -4).Final AnswerThe integer pairs are boxed{(0, 0)} and boxed{(-2, -4)}.</think>To determine all pairs ((a, b)) of integers such that (a(a - b) = b), we start by expanding and rearranging the equation:[a(a - b) = b implies a^2 - ab = b implies a^2 - ab - b = 0 implies a^2 = b(a + 1)]This implies that (a + 1) must divide (a^2). Since (a) and (a + 1) are coprime, (a + 1) must divide 1. Therefore, (a + 1) can only be (pm 1):- If (a + 1 = 1), then (a = 0). Substituting (a = 0) into the equation gives (b = 0).- If (a + 1 = -1), then (a = -2). Substituting (a = -2) into the equation gives (b = -4).We verify these solutions:- For ((0, 0)), the equation becomes (0(0 - 0) = 0), which is true.- For ((-2, -4)), the equation becomes (-2(-2 - (-4)) = -4), which simplifies to (-2 cdot 2 = -4), also true.Other values of (a) are checked and found not to yield integer solutions for (b). Thus, the only integer pairs are:[boxed{(0, 0)} quad text{and} quad boxed{(-2, -4)}]]

🔑:## First solution.Let's write the equation in the form a^{2}=b(a+1).The number a+1 is relatively prime to a, and it must be a divisor of a^{2}.4 pointsTherefore, it must be a+1=1 or a+1=-1. 3 pointsIn the first case, a=0 and b=0, and in the second case, a=-2 and b=-4.

🧠:[<think>Okay, let's try to solve this problem. So, we have two quadratic equations: the first one is (2x^2 - 18x + 36 = 0) and the second one is (x^2 - 20x + 75 = 0). The roots of these equations are the lengths of four segments. Then we need to determine whether these segments can form a quadrilateral with a perimeter of either 38 or 29, or if they cannot form a quadrilateral at all. The options are A, B, C, or D (none of the above). First, I need to find the roots of both equations. Let me start with the first equation: (2x^2 - 18x + 36 = 0). To make it simpler, maybe I can divide all terms by 2 to reduce the coefficients. That would give me (x^2 - 9x + 18 = 0). Now, I can factor this quadratic equation. Let's see, two numbers that multiply to 18 and add up to 9. Hmm, 6 and 3. Yes, that works. So, the factored form is ((x - 6)(x - 3) = 0). Therefore, the roots are 6 and 3. So, from the first equation, the segment lengths are 6 and 3.Now, moving on to the second equation: (x^2 - 20x + 75 = 0). Again, let's try factoring. We need two numbers that multiply to 75 and add up to -20. Since the middle term is negative and the constant term is positive, both numbers should be negative. Let's see, 75 factors into 5 and 15, but 5 + 15 is 20. So, if we make them both negative, then yes, that would work. So, the equation factors as ((x - 5)(x - 15) = 0). Wait, hold on. If I factor (x^2 - 20x + 75), then I need two numbers that multiply to +75 and add up to -20. So, both numbers should be negative. So, -5 and -15. Therefore, the factors are ((x - 5)(x - 15)), which would expand to (x^2 - 20x + 75). Correct. So, the roots are 5 and 15. Therefore, the segment lengths from the second equation are 5 and 15.So now, combining the roots from both equations, the four segment lengths are 3, 6, 5, and 15. Let me list them again to make sure I didn't make a mistake. First equation: 3 and 6, second equation: 5 and 15. Yep, that seems right.Next, the problem states that these four segments can form a quadrilateral. Wait, but quadrilaterals have four sides, right? So, if these four segments are the sides, then they need to satisfy the triangle inequality extended to quadrilaterals. The rule is that the sum of any three sides must be greater than the fourth side. So, for four sides a, b, c, d, the following must hold:1. a + b + c > d2. a + b + d > c3. a + c + d > b4. b + c + d > aIf all these conditions are met, then the four segments can form a quadrilateral. Otherwise, they cannot.So, let's check these conditions with the given lengths: 3, 5, 6, and 15. Let's order them from smallest to largest: 3, 5, 6, 15. The largest side is 15. So, the sum of the other three sides is 3 + 5 + 6 = 14. Now, 14 is less than 15. So, 14 < 15. Therefore, 3 + 5 + 6 = 14 is not greater than 15. Therefore, the first condition fails. Therefore, these four segments cannot form a quadrilateral because the sum of the three smaller sides is not greater than the largest side.Wait, but hold on. The problem mentions that the four roots are the lengths of four segments. So, perhaps they can be arranged in different ways? Wait, but in a quadrilateral, the order of sides doesn't matter. The only requirement is that the sum of any three sides must be greater than the fourth. So, regardless of how you arrange them, if the sum of the three smallest sides is not greater than the largest side, then it's impossible to form a quadrilateral. So in this case, the three smaller sides add up to 3 + 5 + 6 = 14, which is less than 15. Therefore, they cannot form a quadrilateral. So, the answer would be option (C) cannot form a quadrilateral. However, the options given are (A) can form a quadrilateral with perimeter 38, (B) can form with perimeter 29, (C) cannot form, (D) none of the above.But wait, let me check the perimeters just in case. The perimeter would be the sum of all four sides: 3 + 5 + 6 + 15 = 29. Wait, that's 3 + 5 is 8, plus 6 is 14, plus 15 is 29. So perimeter is 29. But the problem is, even though the perimeter is 29, the sides cannot form a quadrilateral because of the triangle inequality. So, even though the perimeter is 29, the quadrilateral can't exist. Therefore, option (B) says they can form a quadrilateral with perimeter 29, but in reality, even though the perimeter is 29, the quadrilateral can't be formed. Therefore, (B) is incorrect. Similarly, (A) mentions perimeter 38, which is not the case here. Therefore, the correct answer is (C). However, let me check again to make sure I didn't miscalculate anything.Wait, perimeter: 3 + 5 + 6 + 15. Let's compute it again. 3 + 5 is 8, 8 + 6 is 14, 14 + 15 is 29. Yes, that's correct. So perimeter is 29, but since the sum of the three smaller sides is 14, which is less than 15, they can't form a quadrilateral. Therefore, the answer should be (C). But wait, let's check the problem statement again. The options are:(A) can form a quadrilateral with a perimeter of 38.(B) can form a quadrilateral with a perimeter of 29.(C) cannot form a quadrilateral.(D) none of the above.So, according to my reasoning, even though the perimeter would be 29, the quadrilateral cannot be formed because the triangle inequality is violated. Therefore, (C) is the correct answer. However, maybe I need to check again if the problem is asking something different. For example, maybe the roots are the sides, but maybe they can be used as sides in some other way? Wait, no. If you have four sides, regardless of their arrangement, if the sum of the three smallest sides is not greater than the largest side, then it's impossible. Therefore, (C) is the answer.But let me cross-verify. Suppose someone thinks that maybe the segments can be rearranged or something. But no, the four sides are fixed. The four lengths are 3, 5, 6, 15. The largest side is 15, which is greater than the sum of the other three. Therefore, no matter how you arrange them, you can't form a quadrilateral. Therefore, (C) is correct. So, the answer is (C). Therefore, the answer is boxed{C}.Wait, but hold on. Let me check the problem statement again. It says, "the roots of the equations... are the lengths of four segments". So, the first equation has two roots: 3 and 6. The second equation has two roots: 5 and 15. So, total four segments: 3, 6, 5, 15. Correct. So, then, those are the four lengths. Therefore, those are the sides. So, as per quadrilateral inequality, the sum of any three sides must be greater than the fourth. As we saw, 3 + 5 + 6 = 14 < 15. Therefore, inequality fails. Hence, cannot form a quadrilateral. Therefore, answer is (C). However, the options include (C) "cannot form a quadrilateral", which is correct. Therefore, the answer should be (C). But wait, the original problem is from a Chinese competition, and sometimes these problems can be tricky. Let me check again. Maybe I made a mistake in calculating the roots?First equation: (2x^2 - 18x + 36 = 0). Divided by 2: (x^2 - 9x + 18 = 0). Factored into (x - 3)(x - 6) = 0. So roots at 3 and 6. Correct.Second equation: (x^2 - 20x + 75 = 0). Factored into (x - 5)(x - 15) = 0. Roots at 5 and 15. Correct. So, 3, 6, 5, 15. Sum of all four: 3 + 6 + 5 + 15 = 29. Correct. The problem says, can they form a quadrilateral with perimeter 38 or 29. Since 29 is the actual perimeter, but since the quadrilateral can't be formed, option (B) is incorrect. Therefore, the answer is (C). So, boxed{C} is the correct answer. Wait, but hold on. Let me check if there is a different interpretation. For example, maybe the four roots are considered as possible sides, but not necessarily all used. But the problem says "the roots... are the lengths of four segments", so that would mean we have four segments, each being a root. So, 3, 6, 5, 15. So, all four must be used. Therefore, the perimeter is 29, but they cannot form a quadrilateral. So, (C) is correct. Therefore, the answer is (C). Therefore, I think the answer is (C). Final Answerboxed{C}</think>To determine whether the roots of the equations (2x^2 - 18x + 36 = 0) and (x^2 - 20x + 75 = 0) can form a quadrilateral, we first find the roots of each equation.For the first equation (2x^2 - 18x + 36 = 0):- Dividing by 2 gives (x^2 - 9x + 18 = 0).- Factoring, we get ((x - 3)(x - 6) = 0).- The roots are (3) and (6).For the second equation (x^2 - 20x + 75 = 0):- Factoring, we get ((x - 5)(x - 15) = 0).- The roots are (5) and (15).The four segments are (3), (6), (5), and (15). To check if they can form a quadrilateral, we use the quadrilateral inequality: the sum of any three sides must be greater than the fourth side.- The largest side is (15).- The sum of the other three sides is (3 + 5 + 6 = 14).Since (14 < 15), the quadrilateral inequality is violated. Therefore, the segments cannot form a quadrilateral. The perimeter of the segments is (3 + 5 + 6 + 15 = 29), but since the quadrilateral cannot be formed, the correct answer is:(boxed{C}),<think>Alright, let's tackle this problem step by step. The question involves two quadratic equations, their roots as the lengths of four segments, and determining whether these segments can form a quadrilateral with specific perimeters or not. The options given are A, B, C, or D. Let's start by understanding what's being asked here.First, I need to find the roots of both quadratic equations because those roots represent the lengths of the four segments. Once I have the lengths, I need to check if they can form a quadrilateral. The key thing to remember here is the triangle inequality extended to quadrilaterals. For four lengths to form a quadrilateral, the sum of any three sides must be greater than the fourth side. If all four sides satisfy this condition, then they can form a quadrilateral. If not, then they cannot. The options given mention specific perimeters (38 and 29), but I also need to check if such perimeters are possible given the side lengths.Let me start by solving both quadratic equations.First equation: 2x² - 18x + 36 = 0I can simplify this equation by dividing all terms by 2 to make it easier:x² - 9x + 18 = 0Now, let's factor this quadratic equation. Looking for two numbers that multiply to 18 and add up to -9. Those numbers are -6 and -3.So, (x - 6)(x - 3) = 0Therefore, the roots are x = 6 and x = 3. But wait, since this is a quadratic equation, each root is repeated once? Wait, no. The equation is quadratic, so there are two roots. However, if there was a repeated root, it would be a double root. But in this case, the roots are distinct: 6 and 3. So, the first equation gives us two lengths: 3 and 6.Wait, hold on. Wait, the original equation is 2x² - 18x + 36 = 0. After dividing by 2, we get x² - 9x + 18 = 0, which factors to (x - 3)(x - 6) = 0. So yes, the roots are 3 and 6. So two segments from the first equation: 3 and 6.Second equation: x² - 20x + 75 = 0Let's factor this quadratic. Looking for two numbers that multiply to 75 and add up to -20. Hmm, 75 factors into 1 and 75, 3 and 25, 5 and 15. The pair 5 and 15 add up to 20. Since the middle term is -20x, both numbers need to be negative. So:(x - 5)(x - 15) = 0Therefore, the roots are x = 5 and x = 15. So the second equation gives us two segments: 5 and 15.Therefore, combining both equations, we have four segments: 3, 6, 5, and 15.So now, the four lengths are 3, 5, 6, and 15. The problem states that these four segments are the lengths, and we need to check if they can form a quadrilateral. The options given are about the perimeter, but first, we need to check the quadrilateral inequality.Wait, quadrilateral inequality: For any quadrilateral, the sum of any three sides must be greater than the fourth side. Let's check each combination.So, the sides are 3, 5, 6, 15. Let's check if the sum of the three smaller sides is greater than the largest side.Sum of 3, 5, and 6: 3 + 5 + 6 = 14. The largest side is 15. 14 is not greater than 15. Therefore, 14 < 15. So this fails the quadrilateral inequality. Therefore, these four segments cannot form a quadrilateral. Therefore, the answer would be option C: cannot form a quadrilateral.But wait, let's confirm. Maybe I made a mistake here. Let me double-check the sums. The largest side is 15, so we need to check if 3 + 5 + 6 > 15. 3 + 5 + 6 = 14, which is less than 15. So indeed, the sum of the three smaller sides is not greater than the largest side. Therefore, a quadrilateral cannot be formed. So option C is correct.However, the problem gives options A and B as possible perimeters, but if they cannot form a quadrilateral, then those perimeters are irrelevant. Therefore, the correct answer is C: cannot form a quadrilateral.But wait, the options mention perimeters of 38 and 29. Let me calculate the total sum of all sides. 3 + 5 + 6 + 15 = 29. So the perimeter would be 29 if they could form a quadrilateral, but since they can't, then option B says "can form a quadrilateral with a perimeter of 29," but since they can't form a quadrilateral at all, option C is correct. So the answer is C.But let me check again. The sum of the four sides is 29, but since 3 + 5 + 6 = 14 < 15, the largest side is too long. So even though the perimeter is 29, they can't form a quadrilateral. Therefore, the correct answer is C.But wait, I just want to confirm if quadrilateral inequality is indeed about the sum of any three sides. Let me recall. For a quadrilateral with sides a, b, c, d, where d is the longest side, the necessary and sufficient condition is that a + b + c > d. If that's not satisfied, then it can't form a quadrilateral. So yes, that's the rule. So since 3 + 5 + 6 = 14 < 15, it fails. Therefore, even though the total perimeter is 29, they cannot form a quadrilateral, so answer C.But let's check the options again:(A) can form a quadrilateral with a perimeter of 38.(B) can form a quadrilateral with a perimeter of 29.(C) cannot form a quadrilateral.(D) none of the above.But the total perimeter of the four sides is 3 + 5 + 6 + 15 = 29. So option B is stating that they can form a quadrilateral with that perimeter. However, even though the perimeter is 29, they cannot form a quadrilateral at all. Therefore, option C is correct.But wait, the problem might be tricky here. Maybe the problem is considering that even if the four sides cannot form a quadrilateral, but if you use different combinations, but no. The problem says "the roots of the equations... are the lengths of four segments," so all four roots are the lengths. Therefore, the four segments are 3,5,6,15, and we must use all four to form a quadrilateral, but they can't. Therefore, the answer is C.But hold on, another thought: sometimes, problems might consider that you can use the sides in a different way, but in reality, the quadrilateral must have four sides, each of those lengths. So if the four lengths cannot satisfy the quadrilateral inequality, then it's impossible. Therefore, regardless of the perimeter, if the four sides can't form a quadrilateral, then C is correct.Therefore, the answer is C.But let me check again if there is any possible way to arrange the sides. For example, even if one side is very long, but maybe by adjusting the angles... Wait, no. In Euclidean geometry, the quadrilateral inequality is a strict requirement. If the sum of the three smaller sides is not greater than the largest side, then it's impossible to form a quadrilateral, no matter how you arrange them. Because the sides are rigid; you can't bend them to make up for the lack of length. So, for example, imagine trying to connect three short sticks end-to-end; if their total length is less than the fourth stick, you can't close the shape. So yes, it's impossible.Therefore, the correct answer is C. So the answer is option C.Final Answerboxed{C}</think>To determine if the roots of the equations (2x^2 - 18x + 36 = 0) and (x^2 - 20x + 75 = 0) can form a quadrilateral, we first find the roots of each equation.1. Solving (2x^2 - 18x + 36 = 0): - Divide by 2: (x^2 - 9x + 18 = 0) - Factor: ((x - 3)(x - 6) = 0) - Roots: (x = 3) and (x = 6)2. Solving (x^2 - 20x + 75 = 0): - Factor: ((x - 5)(x - 15) = 0) - Roots: (x = 5) and (x = 15)The lengths of the four segments are 3, 5, 6, and 15. To check if these can form a quadrilateral, we use the quadrilateral inequality: the sum of any three sides must be greater than the fourth side.- The largest side is 15. Checking the sum of the other three sides: (3 + 5 + 6 = 14)- Since (14 < 15), the segments fail the quadrilateral inequality.Thus, the segments cannot form a quadrilateral. The correct answer is (boxed{C}).]

🔑:[Solution] It is not difficult to find that the roots of the two equations are 3, 6 and 5, 15, respectively.By 3+5+6=14<15,that is, one of the line segments is longer than the sum of the other three line segments, which clearly cannot form a quadrilateral. Therefore, the answer is (C).

🧠:[<think>Okay, let's see. The problem is asking me to convert the number 0.000000032 meters into scientific notation. Hmm, I remember that scientific notation is a way to express very large or very small numbers in a more compact form. It usually involves writing the number as a product of a number between 1 and 10 and a power of 10.So, first, I need to recall how scientific notation works. The general form is a × 10^b, where a is a number greater than or equal to 1 and less than 10, and b is an integer. For numbers less than 1, like this one, the exponent b will be negative because you're moving the decimal point to the right to make the number between 1 and 10.Let me look at the given number: 0.000000032. That's 32 billionths, right? So there are 8 zeros after the decimal point before the 3. Wait, let's count the places. Starting from the decimal point:1st position: 02nd: 03rd: 04th: 05th: 06th: 07th: 08th: 09th: 310th: 2Wait, so the number is 0.000000032. Let me count the zeros. After the decimal, there are 8 zeros and then 32. So that's 8 zeros between the decimal and the first non-zero digit, which is 3. So to convert this to scientific notation, I need to move the decimal point to the right after the first non-zero digit. In this case, the first non-zero digit is 3. So moving the decimal point 8 places to the right gives me 3.2 × 10^-8?Wait, let me check. If I start with 0.000000032, moving the decimal point 8 places to the right would be:1. Move to the 1st place: 0.00000003.2 (Wait, but decimal places don't work that way. Let me think again.)Alternatively, every time I move the decimal point to the right by one place, the number becomes 10 times larger. So starting at 0.000000032:Move decimal 1 place: 0.00000032 (×10)Move 2 places: 0.0000032 (×10^2)Move 3 places: 0.000032 (×10^3)Move 4 places: 0.00032 (×10^4)Move 5 places: 0.0032 (×10^5)Move 6 places: 0.032 (×10^6)Move 7 places: 0.32 (×10^7)Move 8 places: 3.2 (×10^8)Ah, so moving the decimal 8 places to the right gives me 3.2. Therefore, to get from 3.2 back to the original number, I have to divide by 10^8, which is the same as multiplying by 10^-8. So the scientific notation is 3.2 × 10^-8.Wait, but let me verify with the original number. 3.2 × 10^-8 would be 0.000000032, which matches the given number. So that's correct. But let me check another way.Another way to think about it is that each zero after the decimal represents a negative power of 10. For example, 0.1 is 10^-1, 0.01 is 10^-2, and so on. So 0.000000032 is 32 × 10^-9? Because after the decimal, there are 9 digits: 8 zeros and then 32. So 32 × 10^-9. But 32 isn't between 1 and 10, so we need to adjust that. 32 × 10^-9 can be written as 3.2 × 10^1 × 10^-9 = 3.2 × 10^-8. Yes, that gives the same result.So either way, whether moving the decimal or factoring the number, we end up with 3.2 × 10^-8. Therefore, the scientific notation for 0.000000032 is 3.2 × 10^-8 meters.But just to be thorough, let me double-check. Let's take 3.2 × 10^-8 and convert it back to standard form. 10^-8 is 0.00000001. Multiplying that by 3.2 gives 0.000000032, which is exactly the number given. Perfect, that matches. So there's no mistake here.I think that's solid. So the answer should be 3.2 × 10^-8. Let me confirm once again with the method of counting decimal places. The original number has 8 zeros after the decimal before the 3. So the exponent is -8 (since we moved the decimal 8 places to the right), and the coefficient is 3.2. Yep, that's correct.Therefore, the number 0.000000032 in scientific notation is 3.2 × 10^-8.Final AnswerThe number (0.000000032) in scientific notation is boxed{3.2 times 10^{-8}}.</think>To express the number (0.000000032) in scientific notation, we need to convert it into a product of a number between 1 and 10 and a power of 10. 1. Identify the non-zero digits: The number (0.000000032) has the non-zero digits 3 and 2.2. Move the decimal point: Move the decimal point 8 places to the right to get the number 3.2. This makes the number 3.2, which is between 1 and 10.3. Determine the exponent: Since we moved the decimal point 8 places to the right, the exponent for 10 will be (-8).Thus, the number (0.000000032) in scientific notation is (3.2 times 10^{-8}).[boxed{3.2 times 10^{-8}}],<think>Okay, so I need to convert the number 0.000000032 meters into scientific notation. Hmm, let me recall how scientific notation works. From what I remember, scientific notation is a way to express very large or very small numbers in a more compact form. It's usually written as a number between 1 and 10 multiplied by a power of 10. The general form is a × 10^b, where 1 ≤ a < 10 and b is an integer. So for numbers less than 1, like this one, the exponent will be negative. That makes sense because moving the decimal point to the right increases the exponent by 1 each time, but since the original number is less than 1, we need negative exponents to represent how many places we moved the decimal to the right.Let me start with the given number: 0.000000032. I need to move the decimal point to the right until I get a number between 1 and 10. Let's count how many places I move the decimal. Starting from the decimal point, the first non-zero digit is 3. Let me count the zeros after the decimal and before the 3. There's 0., then 7 zeros: 0.000000032. So after the decimal, there are seven zeros and then 32. Wait, actually, let's count again. Let's write it out step by step. Original number: 0.000000032Breakdown:1st digit after decimal: 02nd: 03rd: 04th: 05th: 06th: 07th: 08th: 39th: 2So the decimal is at the start, before any digits. To get to between 3.2, which is between 1 and 10, I need to move the decimal point 8 places to the right. Because moving it 8 places would land between the 3 and the 2, making it 3.2.Wait, let me confirm. Starting at 0.000000032, moving the decimal 1 place right: 0.000000032 becomes 0.00000032 (but that's still less than 1). Moving 2 places: 0.0000032, still less than 1. 3 places: 0.000032, 4 places: 0.00032, 5 places: 0.0032, 6 places: 0.032, 7 places: 0.32, 8 places: 3.2. Yes, that's correct. So moving 8 places to the right gives 3.2.Since we moved the decimal 8 places to the right, the exponent will be -8. Because moving to the right for numbers less than 1 gives a negative exponent. So the scientific notation is 3.2 × 10^-8.Wait, let me check if that's accurate. Let's test multiplying 3.2 × 10^-8. That would be 3.2 divided by 10^8, which is 3.2 ÷ 100,000,000. Let's compute that. 3.2 divided by 100 million. 100 million is 10^8. So 3.2 / 10^8 is 0.000000032, which matches the original number. Perfect.But wait, sometimes people might make a mistake in counting the number of decimal places. Let me recount. Original number is 0.000000032. So after the decimal, how many digits are there? Let's see: 0.0 0 0 0 0 0 0 3 2. So from decimal to the first non-zero digit (the 3), there are 8 digits. But since the decimal is moved 8 places to get to 3.2, the exponent is -8. So yes, 8 places. Alternatively, another way to think about it is that each zero after the decimal is a place, and each non-zero digit is a place. So in 0.000000032, there are 7 zeros after the decimal before the 3. Then the 3 is the 8th digit, and the 2 is the 9th. So moving the decimal from the start to after the 3 would require moving 8 places. Therefore, 3.2 × 10^-8 meters. Let me check with another example. If I take a number like 0.0034, which is 3.4 × 10^-3. Here, we moved the decimal 3 places. Similarly, 0.000000032 would be 3.2 × 10^-8. Yep, that's consistent.So the answer should be 3.2 × 10^-8. Let me just confirm once more. If I write 3.2e-8 in decimal form, that's 0.000000032. Yes. That's correct. So no mistakes here.Final AnswerThe number (0.000000032) in scientific notation is boxed{3.2 times 10^{-8}}.</think>To convert the number (0.000000032) into scientific notation, we need to express it in the form (a times 10^b) where (1 leq a < 10) and (b) is an integer.1. Start with the given number: (0.000000032).2. Move the decimal point 8 places to the right to get a number between 1 and 10. This results in (3.2).3. Since we moved the decimal point 8 places to the right, the exponent (b) will be (-8).Thus, the number (0.000000032) in scientific notation is (3.2 times 10^{-8}).[boxed{3.2 times 10^{-8}}]]

🔑:To express the number 0.000000032 in scientific notation, we follow the steps below:1. Identify the significant figures in the number, which are 3 and 2.2. Move the decimal point in the number 0.000000032 so that it is after the first significant figure, which gives us 3.2.3. Count the number of places the decimal point has been moved. In this case, it has been moved 8 places to the right.4. Write the number in the form of a times 10^n, where a is the number obtained in step 2, and n is the negative of the number of places the decimal point has been moved because we moved it to the right.Therefore, 0.000000032 can be expressed in scientific notation as:0.000000032 = 3.2 times 10^{-8}So, the final answer is boxed{3.2 times 10^{-8}}.