🧠:[<think>Okay, let me try to figure out this problem. So, we have two sets, A and B. A is {2, m} and B is {1, m²}. The union of A and B is {1, 2, 3, 9}. We need to find the real number m. Hmm, alright.First, let me recall what a union of sets means. The union of two sets A and B is the set containing all elements that are in A, in B, or in both. So, A ∪ B should have all elements from both sets without repetition. Given that A ∪ B is {1, 2, 3, 9}, that means all the elements from A and B combined must be exactly these numbers: 1, 2, 3, and 9.So, let's list the elements of A and B. Set A has 2 and m. Set B has 1 and m². When we take their union, we get 1, 2, m, and m². But according to the problem, this union is exactly {1, 2, 3, 9}. Therefore, the elements m and m² must correspond to the numbers 3 and 9 in some order. Because the union only has 1, 2, 3, 9, there are no other elements allowed. So m must be either 3 or 9, but we need to check if that's possible.Wait, hold on. If m is 3, then m² would be 9. Let's check if that works. Then set A would be {2, 3} and set B would be {1, 9}. Their union would indeed be {1, 2, 3, 9}, which matches the given union. That seems okay. Now, what if m is 9? Then m² would be 81. But 81 isn't in the union set {1, 2, 3, 9}, so that doesn't work. So m can't be 9. So the only possible value is 3? Wait, but let me make sure there's no other possibility.Another thought: Maybe m is a number that's not 3 or 9, but when squared becomes 3 or 9. For example, if m is sqrt(3), then m² is 3. Let's test that. If m is sqrt(3), then set A is {2, sqrt(3)} and set B is {1, 3}. Then the union would be {1, 2, sqrt(3), 3}, but the problem states the union is {1, 2, 3, 9}. So sqrt(3) is approximately 1.732, which is not in {1, 2, 3, 9}, so that doesn't work. Similarly, if m is sqrt(9), which is 3, but that's already considered. If m is -3, then m² is 9. Let's check that. If m is -3, then set A is {2, -3} and set B is {1, 9}. The union would be { -3, 1, 2, 9}. But the problem says the union is {1, 2, 3, 9}, so -3 is not in the union. Therefore, m cannot be -3. So that's invalid.Wait, the problem says m is a real number. So negative numbers are allowed? But in the union, we have 3, so if m is -3, then m² is 9, but -3 would need to be in the union. But since the union is {1, 2, 3, 9}, and -3 is not there, that can't be. So m must be positive? Maybe. Let me see.Alternatively, maybe m is 3. If m is 3, then m² is 9, which works. So set A is {2, 3}, set B is {1, 9}, union is {1, 2, 3, 9}. Perfect. That matches. So that works. Is there any other possibility? Let's think. Suppose m is a number that's not 3 or 9 but such that both m and m² are in the union set. The union set is {1, 2, 3, 9}. So m must be one of these numbers: 1, 2, 3, 9, or another number whose square is among these. Let's check:If m is 1: Then set A is {2, 1}, set B is {1, 1} which is just {1}. Then the union is {1, 2}, which is not equal to {1, 2, 3, 9}. So m can't be 1.If m is 2: Then set A is {2, 2} which is {2}, set B is {1, 4}. Then union is {1, 2, 4}, which is not equal to {1, 2, 3, 9}. So m can't be 2.If m is 3: As before, works.If m is 9: Then m² is 81, which is not in the union set. So invalid.What if m is sqrt(9) which is 3, already considered. What about m being 3/2? Then m squared is 9/4, which is 2.25, not in the union set. So that's invalid.Alternatively, m is cube root of 9 or something, but m squared would be 9^(2/3), which is 3√9², which is 3*3=9. Wait, 9^(2/3) is (9^(1/3))^2. The cube root of 9 is approximately 2.08, and squaring that is approximately 4.33, which is not 3 or 9. So that's not helpful.Wait, but perhaps m is 3. Then m squared is 9. So that's straightforward.Alternatively, if m squared is 3, then m is sqrt(3). But as before, sqrt(3) is not in the union set. The union set must include all elements of A and B, so if m is sqrt(3), then A is {2, sqrt(3)} and B is {1, 3}. Then the union is {1, 2, sqrt(3), 3}, which includes sqrt(3), which is not in the given union {1, 2, 3, 9}. Therefore, that's invalid.Similarly, if m squared is 1, then m is 1 or -1. Then set A would be {2, 1} or {2, -1}. But then the union would include 1 and 2, and -1 if m is -1, which is not in the given union. So that's not valid.If m squared is 2, then m is sqrt(2) or -sqrt(2). Then the union would include sqrt(2) and 2, but sqrt(2) isn't in the given union set. So that's invalid.If m squared is 9, then m is 3 or -3. As before, m=3 works, m=-3 leads to set A being {2, -3} which introduces -3 not in the union set, so invalid.Therefore, the only valid possibility is m=3. Let me double-check:A = {2, 3}B = {1, 9}A ∪ B = {1, 2, 3, 9}Yes, that's exactly the union given. So m must be 3.I think that's the only solution. Let me confirm there are no other numbers. Suppose m is a real number not in {1, 2, 3, 9}, but m and m² are in {1, 2, 3, 9}. So m must be such that both m and m² are in {1, 2, 3, 9}. Let's list possible pairs:Looking for m where m is in {1, 2, 3, 9} and m² is also in {1, 2, 3, 9}. Let's check each element:1: m=1, m²=1. Then A={2,1}, B={1,1}. Union is {1,2}, not desired.2: m=2, m²=4. Union would be {1,2,4}, not desired.3: m=3, m²=9. Union is {1,2,3,9}, which is desired.9: m=9, m²=81. Union would include 81, not desired.Therefore, only m=3 satisfies the condition that both m and m² are in the union set {1,2,3,9}. Therefore, m must be 3.Final AnswerThe real number ( m ) is boxed{3}.</think>Given the sets ( A = {2, m} ) and ( B = {1, m^2} ), and the union ( A cup B = {1, 2, 3, 9} ), we need to find the real number ( m ).1. The union ( A cup B ) contains all elements from both sets ( A ) and ( B ). Therefore, the elements ( m ) and ( m^2 ) must be among the numbers 1, 2, 3, and 9.2. We check possible values for ( m ): - If ( m = 3 ), then ( m^2 = 9 ). The sets become ( A = {2, 3} ) and ( B = {1, 9} ), and their union is ( {1, 2, 3, 9} ), which matches the given union. - Other values such as ( m = 1 ), ( m = 2 ), ( m = 9 ), or negative numbers like ( m = -3 ) do not satisfy the condition because they either introduce elements not in the union or do not match the required elements.Thus, the only valid solution is ( m = 3 ).[boxed{3}],<think>Okay, let's see. I have this problem here where I need to find the real number m given the sets A and B. The problem states that A is {2, m} and B is {1, m²}. The union of A and B is {1, 2, 3, 9}. So, I need to figure out what m is such that when I combine sets A and B, I get exactly those four elements: 1, 2, 3, and 9. First, let me recall what a union of two sets means. The union of sets A and B, denoted as A ∪ B, is the set of all elements that are in A, in B, or in both. So, combining A and B without repetition. The problem says that this union is {1, 2, 3, 9}. Therefore, the elements of A and B together must be exactly these four numbers. Given that A is {2, m} and B is {1, m²}, when we take their union, we should get 1, 2, m, and m². Because the union would include all unique elements from both sets. But according to the problem, the union is {1, 2, 3, 9}. So that means that the elements m and m² must correspond to 3 and 9 in some order. Wait, let's break that down. The union of A and B is given as {1, 2, 3, 9}. So the elements 1 and 2 must come from set B and set A respectively. Set A has 2 and m, set B has 1 and m². So 2 is in A, 1 is in B. Then m and m² must be the remaining elements, which are 3 and 9. Therefore, m and m² must be 3 and 9. But which one is which? Is m 3 and m² 9, or is m 9 and m² 81? Wait, but 81 isn't in the union. So m can't be 9 because then m² would be 81, which isn't part of the given union set. So that can't be. Therefore, m must be 3, because then m² would be 9, which is in the union. Let me check that.If m = 3, then set A becomes {2, 3} and set B becomes {1, 9}. Then, the union of A and B would be {1, 2, 3, 9}, which matches exactly the given union. So that works. But wait, let me make sure there's no other possible value for m. Suppose m² is 3 and m is √3 or -√3. Then, m² would be 3, which would need to be in the union. But if m is √3, then set A would be {2, √3} and set B would be {1, 3}. Then their union would be {1, 2, √3, 3}, but √3 is approximately 1.732, which is not equal to 9. So the union would have √3 instead of 9, which doesn't match. Similarly, if m is -√3, same issue. So m² is 3, which is okay, but then m is not an integer, and the union would have an irrational number, which isn't in the given union set {1, 2, 3, 9}. Therefore, this case is invalid.Alternatively, could m be -3? Let's check. If m = -3, then set A would be {2, -3} and set B would be {1, 9} (since (-3)² is 9). Then the union would be {1, 2, -3, 9}. But the given union is {1, 2, 3, 9}. Here, we have -3 instead of 3, which is different. So unless the problem allows negative numbers, but the union set includes 3, not -3. Therefore, m can't be -3.Another possibility: maybe m is 9? Then m² is 81. But 81 isn't in the union set, so that doesn't work. Similarly, m could be 1, but then m² is 1, and then set A would be {2, 1}, set B would be {1, 1}, which simplifies to {1, 2}, and their union would be {1, 2}, which is not the given union. So m can't be 1.What if m is 2? Then m² is 4. Then set A is {2, 2} which is just {2}, and set B is {1, 4}. The union would be {1, 2, 4}, which is not the required union. So that's out.Alternatively, maybe m is 3 and m² is 9. Which works as we saw earlier. Let's confirm again. If m=3, A={2,3}, B={1,9}, so A∪B={1,2,3,9}, which is exactly what's given. So that works.But wait, let's check if there's another value. Suppose m is such that m is 9 and m² is 81. But 81 is not in the union, so that's invalid. If m is something else, like 0, then m² is 0. Then union would have 0, which isn't in the given set. So no.Wait, but maybe m is sqrt(9) which is 3, but that's the same as before. Or m is cube root of something? But in the union, there's 3 and 9, so maybe m is 3. Since 3 squared is 9, so that works. Alternatively, m could be -3, but as before, that introduces -3 into the union, which is not present.Therefore, the only real number m that satisfies the condition is 3. Let me just check once again. If m=3, then A={2,3}, B={1,9}, union is {1,2,3,9}, which matches. Any other m would either result in a union with elements not in the given set or missing elements. Therefore, m must be 3.Final AnswerThe real number ( m ) is boxed{3}.</think>Given the sets ( A = {2, m} ) and ( B = {1, m^2} ), we need to find the real number ( m ) such that ( A cup B = {1, 2, 3, 9} ).1. The union of sets ( A ) and ( B ) must include all elements from both sets without repetition. 2. The given union is ( {1, 2, 3, 9} ). Therefore, the elements ( m ) and ( m^2 ) must be 3 and 9, respectively.3. We check if ( m = 3 ) satisfies the condition: - If ( m = 3 ), then ( A = {2, 3} ) and ( B = {1, 9} ). - The union ( A cup B = {1, 2, 3, 9} ), which matches the given union set.4. Other values for ( m ) (such as -3, 9, or non-integer values) either result in elements not present in the union set or do not satisfy the required elements.Thus, the real number ( m ) is (boxed{3}).]

🔑:Since Acup B={1,2,3,9},We have two possible cases:begin{cases}m=3 m^{2}=9end{cases} or begin{cases}m=9 m^{2}=3end{cases},Solving the first case, we get m=3.Solving the second case leads to an inconsistency since the square of a real number cannot be less than the number itself.Hence, the answer is: boxed{m=3}.This problem can be solved using basic set operations. The main focus of this question is to test the understanding of set union and element membership.

🧠:[<think>Okay, so I have this problem here. Let me try to parse it step by step. The question says that vectors a and b are unit vectors, and they are perpendicular to each other. Then there's another vector c that satisfies the condition |c - a - b| = 2. We need to find the range of possible values for |c|. The options are given as A, B, C, D. First, since a and b are unit vectors and perpendicular, they form an orthonormal basis. That might be useful. Let me visualize this. If a and b are perpendicular unit vectors, then they can be considered as the standard basis vectors in 2D space, right? So maybe I can model this problem in 2D to simplify things. Although, the problem doesn't specify the dimension, but since they are perpendicular, it's probably safe to assume they are in 2D. So, let's set up a coordinate system where vector a is along the x-axis, and vector b is along the y-axis. Then, a would be (1, 0) and b would be (0, 1). Then, the vector a + b would be (1, 1). So vector c is such that the distance between c and (1, 1) is 2. So geometrically, c lies on a circle centered at (1, 1) with radius 2. The question is asking for the range of |c|, which is the distance from the origin to the point c. So, essentially, we need to find the minimum and maximum distances from the origin to any point on this circle. Ah, right! The distance from a point to a circle. The formula for the minimum distance is the distance from the origin to the center of the circle minus the radius, and the maximum distance is the distance from the origin to the center plus the radius. Wait, let me confirm that. If we have a circle with center at (h, k) and radius r, then the minimum distance from a point (x0, y0) to the circle is |d - r| where d is the distance from (x0, y0) to (h, k). Wait, no. Actually, if the point is outside the circle, then the minimum distance is d - r, and the maximum distance is d + r. If the point is inside the circle, then the minimum distance is r - d and the maximum is r + d. But in this case, the origin's distance to the center (1,1) is sqrt(1^2 + 1^2) = sqrt(2). The radius of the circle is 2. Since sqrt(2) is approximately 1.414, which is less than 2, the origin is inside the circle. Therefore, the minimum distance from the origin to the circle would be r - d = 2 - sqrt(2), and the maximum would be r + d = 2 + sqrt(2). So, the range is [2 - sqrt(2), 2 + sqrt(2)]. Looking at the options, that's option B. So is that the answer? Wait, but let me make sure I haven't made a mistake here. Let me go through it again. Given that the center of the circle is (1,1), so the distance from the origin to the center is sqrt(1 + 1) = sqrt(2). The radius is 2. Since sqrt(2) ≈ 1.414 is less than 2, the origin is inside the circle. Therefore, the closest point on the circle to the origin would be along the line connecting the origin to the center, but in the opposite direction from the center. Wait, actually, no. If the origin is inside the circle, then the minimum distance from the origin to the circle is the radius minus the distance from the origin to the center, right? Wait, that can't be. Wait, no. If the origin is inside the circle, then the closest distance from the origin to any point on the circle is the distance from the origin to the center minus the radius? Wait, no, I'm confused. Let me think again.Imagine the circle with center C and radius r. If a point P is inside the circle, then the minimum distance from P to the circle is r - |PC|, and the maximum distance is r + |PC|. So if the origin is inside the circle, then the minimum value of |c| (distance from origin to a point on the circle) would be |OC| - r, but wait, no. Wait, actually, the minimum distance from a point inside the circle to the circle is r - |PC|, and the maximum is r + |PC|. Wait, that seems contradictory.Wait, let's take a simple example. Suppose the circle has center at (0,0) and radius 2. Then, the distance from the origin (which is the center) to the circle is 2, so the minimum and maximum would both be 2? No, that doesn't make sense. Wait, no, if the point is the center, then all points on the circle are exactly 2 units away, so the range is [2,2]. But in our problem, the point is not the center.Wait, another example: suppose the circle is centered at (1,0) with radius 1. The origin is at (0,0). The distance from origin to center is 1, which is equal to the radius. So the origin lies on the circle. Therefore, the minimum distance from origin to the circle is 0 (since the origin is on the circle), and the maximum is 2 (the farthest point on the circle from the origin is (2,0)).Another example: suppose the circle is centered at (3,0) with radius 2. The origin is at (0,0). The distance from origin to center is 3, radius is 2. Since 3 > 2, the origin is outside the circle. Then the closest distance is 3 - 2 = 1, and the farthest is 3 + 2 = 5. So generalizing, if the distance from the point to the center is d, and the radius is r:- If d > r (point outside the circle), then minimum distance is d - r, maximum is d + r.- If d <= r (point inside or on the circle), then minimum distance is r - d, maximum is r + d. Wait, no. Wait in the previous example where the center is (1,0) with radius 1, d = 1, which is equal to r. Then the minimum distance is 0 (since the origin is on the circle) and maximum is 2. But according to the formula r - d = 0, which matches the minimum, and r + d = 2, which matches the maximum. If the origin is inside the circle, say center at (0.5, 0), radius 1. Then the distance from origin to center is 0.5. The minimum distance from origin to the circle would be 1 - 0.5 = 0.5, and maximum would be 1 + 0.5 = 1.5. But wait, let's check. The closest point would be in the direction opposite to the center, so (0.5 - 1, 0) = (-0.5, 0), distance 0.5. The farthest point would be in the direction of the center, (0.5 + 1, 0) = (1.5, 0), distance 1.5. So that works.Therefore, the formula is:If the distance from the point to the center is d, then the minimum distance from the point to the circle is |d - r|, and the maximum is d + r. Wait, but when the point is inside, |d - r| = r - d, which gives the minimum distance as r - d, and maximum as r + d. Wait, but when the point is outside, |d - r| = d - r, which is the minimum, and maximum is d + r.Wait, but in the previous example where d = 3, r = 2, min distance is d - r = 1, max is d + r = 5, which is correct.So yes, the general formula is:Minimum distance = |d - r|Wait, but when the point is inside, |d - r| = r - d, which is the minimum distance. When the point is outside, |d - r| = d - r, which is the minimum. Wait, but the maximum distance is always d + r. So perhaps the minimum is |d - r| and maximum is d + r.Wait, but in the case when the point is inside the circle, the maximum distance from the point to any point on the circle is d + r. But in the example where the center is (0.5, 0), radius 1, origin is at (0,0). The distance from origin to center is 0.5, so the maximum distance should be 0.5 + 1 = 1.5, which is correct as the point (1.5, 0) is 1.5 away. The minimum is 1 - 0.5 = 0.5, which is the point (-0.5, 0). So that works.Therefore, in general, regardless of whether the point is inside or outside, the minimum distance from the point to the circle is |d - r|, and the maximum is d + r.Therefore, in our problem, the distance from origin to the center (1,1) is d = sqrt(2). The radius r is 2. Therefore, the minimum |c| is |sqrt(2) - 2| = 2 - sqrt(2), since sqrt(2) is less than 2. The maximum |c| is sqrt(2) + 2. So the range is [2 - sqrt(2), 2 + sqrt(2)], which is option B. That seems correct.Wait, let me verify with coordinates. The circle is centered at (1,1) with radius 2. The origin is (0,0). The closest point on the circle to the origin would be along the line from the origin through the center (1,1). The direction vector from origin to center is (1,1). The unit vector in that direction is (1/√2, 1/√2). The farthest point from the origin on the circle would be in the same direction: center plus radius times the unit vector. So, (1,1) + 2*(1/√2, 1/√2) = (1 + 2/√2, 1 + 2/√2) = (1 + √2, 1 + √2). The distance from origin to this point is sqrt[(1 + √2)^2 + (1 + √2)^2] = sqrt[2*(1 + 2√2 + 2)] = sqrt[2*(3 + 2√2)] = sqrt[6 + 4√2]. Wait, but 2 + sqrt(2) squared is (2 + sqrt(2))^2 = 4 + 4√2 + 2 = 6 + 4√2, which matches. So the maximum distance is 2 + sqrt(2). Similarly, the closest point would be in the opposite direction from the center. The point would be center minus radius times the unit vector in the direction of the center. So (1,1) - 2*(1/√2, 1/√2) = (1 - 2/√2, 1 - 2/√2) = (1 - √2, 1 - √2). The distance from origin to this point is sqrt[(1 - √2)^2 + (1 - √2)^2] = sqrt[2*(1 - 2√2 + 2)] = sqrt[2*(3 - 2√2)] = sqrt[6 - 4√2]. Wait, but (2 - sqrt(2))^2 is 4 - 4√2 + 2 = 6 - 4√2, so sqrt(6 - 4√2) is 2 - sqrt(2). Wait, is that true? Let's check (2 - sqrt(2))^2: 4 - 4√2 + 2 = 6 - 4√2. So sqrt(6 - 4√2) is indeed 2 - sqrt(2). Because (2 - sqrt(2))^2 = 6 - 4√2. Therefore, the minimum distance is 2 - sqrt(2). Therefore, the range of |c| is [2 - sqrt(2), 2 + sqrt(2)], which is option B. That seems correct. Therefore, the answer should be B. But let me just make sure that there isn't a mistake in assuming 2D. The problem didn't specify the dimension, but since a and b are perpendicular unit vectors, they can be in any dimension, but the analysis might be different. Wait, but in higher dimensions, the vector c could be anywhere in space, but given that we have a and b as two perpendicular unit vectors, maybe we can consider the problem in the plane spanned by a and b. Because the vector c is related to a and b via the equation |c - a - b| = 2. If c had components outside the plane of a and b, then the norm |c - a - b| would be sqrt(|c_proj - a - b|^2 + |c_perp|^2), where c_proj is the projection onto the plane and c_perp is the component perpendicular to the plane. For this to equal 2, the minimal |c| would occur when c is in the plane, because any component out of the plane would increase |c|. Wait, no. Wait, |c| is the norm of the entire vector. If c has a component perpendicular to the plane, then |c| would be sqrt(|c_proj|^2 + |c_perp|^2). So if we want to minimize or maximize |c| given that |c - a - b| = 2, then maybe the extrema occur when c is in the plane. Because adding a perpendicular component would require that |c - a - b| = 2, so if c has a component perpendicular, then |c_proj - a - b|^2 + |c_perp|^2 = 4. Therefore, |c_proj - a - b|^2 <= 4, which would mean that |c_proj - a - b| <= 2. But in our problem, |c - a - b| is exactly 2, so if there is a perpendicular component, then |c_proj - a - b| would be less than 2, which might allow for smaller or larger |c|. Hmm, so maybe the range could be different if c is allowed to have components outside the plane. Wait, but the problem doesn't specify the dimension. So if a and b are in 3D space, then c can be anywhere in 3D. So perhaps my previous analysis in 2D is insufficient. Let me consider this in 3D.Let me denote the plane spanned by a and b as the xy-plane, with a along x-axis and b along y-axis. Then, vector a + b is (1,1,0). The vector c can be anywhere in 3D space such that |c - (1,1,0)| = 2. So c lies on a sphere in 3D with center at (1,1,0) and radius 2. The question is to find the range of |c|, which is the distance from the origin (0,0,0) to any point c on this sphere. In 3D, the distance from the origin to the center of the sphere is still sqrt(1^2 + 1^2 + 0^2) = sqrt(2). The radius is 2. The minimum and maximum distances from the origin to the sphere are still sqrt(2) - 2 and sqrt(2) + 2, but since sqrt(2) < 2, the minimum distance is 2 - sqrt(2), and the maximum is 2 + sqrt(2). Wait, but in 3D, the sphere is not a circle, but does this formula still apply? Yes, because for a sphere, the minimum distance from a point to the sphere is the distance from the point to the center minus the radius, if the point is outside the sphere. If the point is inside the sphere, then the minimum distance is the radius minus the distance from the center to the point. Wait, but in 3D, if the origin is inside the sphere, then the minimum distance from the origin to the sphere is the radius minus the distance from center to origin. But the maximum distance is the distance from the origin to the center plus the radius. Wait, hold on. Let's think in terms of 3D geometry. The origin is at (0,0,0). The sphere is centered at (1,1,0) with radius 2. The distance from the origin to the center is sqrt(2), which is approximately 1.414. Since this is less than the radius 2, the origin is inside the sphere. Therefore, the minimum distance from the origin to the sphere is 2 - sqrt(2), and the maximum distance is 2 + sqrt(2). Therefore, the range of |c| is [2 - sqrt(2), 2 + sqrt(2)], same as in the 2D case. Therefore, even if the problem is in 3D or higher dimensions, the answer remains the same. Because regardless of the dimension, the minimal and maximal distances from the origin to the sphere (or circle in 2D) are determined solely by the distance from the origin to the center and the radius. Therefore, the answer is option B. Therefore, I think option B is correct. The other options can be ruled out. Option C is [sqrt(2), 2 sqrt(2)], but sqrt(2) is approximately 1.414, and 2 sqrt(2) is approximately 2.828. But our calculated minimum is 2 - sqrt(2) ≈ 0.585 and maximum is 2 + sqrt(2) ≈ 3.414, which isn't covered by option C. Option D has 3 - 2 sqrt(2) ≈ 3 - 2.828 ≈ 0.172 and 3 + 2 sqrt(2) ≈ 5.828, which is wider than our range. Option A is [1, 1 + sqrt(2)] ≈ [1, 2.414], which is also incorrect. So option B is the correct one.Final Answerboxed{B}</think>Given that (vec{a}) and (vec{b}) are unit vectors and perpendicular to each other, and vector (vec{c}) satisfies (|vec{c} - vec{a} - vec{b}| = 2), we need to find the range of values for (|vec{c}|).1. Orthonormal Basis: Since (vec{a}) and (vec{b}) are perpendicular unit vectors, they form an orthonormal basis. We can consider them as the standard basis vectors in 2D space, (vec{a} = (1, 0)) and (vec{b} = (0, 1)). Thus, (vec{a} + vec{b} = (1, 1)).2. Geometric Interpretation: The vector (vec{c}) lies on a circle (or sphere in 3D) centered at ((1, 1)) with radius 2. We need to find the minimum and maximum distances from the origin ((0, 0)) to any point on this circle/sphere.3. Distance Calculations: - The distance from the origin to the center ((1, 1)) is (sqrt{1^2 + 1^2} = sqrt{2}). - The radius of the circle/sphere is 2.4. Range of Distances: - Since the origin is inside the circle/sphere (as (sqrt{2} approx 1.414 < 2)), the minimum distance from the origin to the circle/sphere is (2 - sqrt{2}). - The maximum distance from the origin to the circle/sphere is (2 + sqrt{2}).5. Conclusion: The range of values for (|vec{c}|) is ([2 - sqrt{2}, 2 + sqrt{2}]), which corresponds to option B.[boxed{B}],<think>Okay, let's see. So the problem is about vectors. We have two unit vectors, a and b, which are perpendicular to each other. Then there's another vector c, which satisfies the condition that the magnitude of c minus a minus b is 2. We need to find the range of possible values for the magnitude of c. The options are given as A, B, C, D. Hmm, alright.First, since a and b are unit vectors and they're perpendicular, they form an orthonormal basis. Maybe I can visualize this in 2D space. Let me recall that if two vectors are perpendicular and of unit length, they can be considered as the standard basis vectors, like i and j. So maybe I can set a as (1,0) and b as (0,1) to simplify things. That should be okay because the problem doesn't specify any particular coordinate system, so choosing a coordinate system where a and b are aligned with the axes might make calculations easier.So, let's let a = (1, 0) and b = (0, 1). Then the vector a + b would be (1, 1). The vector c is such that the distance between c and a + b is 2. Wait, the magnitude of c - (a + b) is 2. So, geometrically, this means that the tip of vector c lies somewhere on a circle (in 2D) with center at the tip of a + b and radius 2. Since we're dealing with vectors in space, but given that a and b are in 2D, maybe c is also in 2D? Or could it be in 3D? The problem doesn't specify, but since a and b are given without any mention of a third dimension, perhaps it's safe to assume they're in 2D. Although, even if they were in 3D, the problem might still be manageable. Let me confirm.In 2D, a + b is (1, 1), which has a magnitude of sqrt(1^2 + 1^2) = sqrt(2). So the vector a + b has length sqrt(2). Then the vector c is such that c is within a circle of radius 2 around a + b. So the question is, what's the minimum and maximum possible length of c?To find the range of |c|, we can think of it as the distance from the origin to points on the circle centered at (1, 1) with radius 2. The minimum distance from the origin to any point on this circle would be the distance from the origin to the center of the circle minus the radius, and the maximum would be the distance to the center plus the radius. Wait, but is that correct?Wait, the distance from the origin to the center of the circle is sqrt(1^2 + 1^2) = sqrt(2). So the minimum |c| would be sqrt(2) - 2, and the maximum would be sqrt(2) + 2. But sqrt(2) is approximately 1.414, so sqrt(2) - 2 is about -0.586, which doesn't make sense because magnitude can't be negative. Therefore, my initial thought is wrong.Hmm, so maybe that approach isn't correct. Wait, in general, for two circles, the minimum distance is the distance between centers minus the sum of radii if one circle is inside the other? Wait, no, actually, if we have a point (the origin) and a circle with center at (1,1) and radius 2, then the minimum distance from the origin to any point on the circle is |distance_between_centers - radius|, and maximum is distance_between_centers + radius. But in this case, since the origin is a point and the circle is around (1,1) with radius 2, the minimum distance from the origin to the circle would be |sqrt(2) - 2|, and the maximum would be sqrt(2) + 2. But sqrt(2) is approximately 1.414, so sqrt(2) - 2 is negative, but distance can't be negative. So the minimum distance is actually zero if the circle overlaps the origin. Wait, but is the origin inside the circle?The circle is centered at (1,1) with radius 2. The distance from the center to the origin is sqrt(2) ≈ 1.414, which is less than the radius 2. Therefore, the origin lies inside the circle. Therefore, the minimum distance from the origin to any point on the circle would be 0, but wait, that can't be right. Wait, if the origin is inside the circle, then the minimum distance from the origin to the circle is radius - distance_between_centers. Wait, no. Wait, actually, when the origin is inside the circle, the closest point on the circle to the origin would be along the line connecting the origin to the center of the circle, extending towards the origin. Wait, but if the origin is inside the circle, then the closest distance from the origin to the circle is the distance from the origin to the center minus the radius? No, that would be negative. Wait, perhaps it's the other way around.Wait, the distance from the origin to the center is sqrt(2). The radius of the circle is 2. Since sqrt(2) < 2, the origin is inside the circle. Therefore, the minimum distance from the origin to the circle is 2 - sqrt(2), and the maximum distance is 2 + sqrt(2). Wait, yes, that makes sense. Because when the origin is inside the circle, the closest point on the circle to the origin would be in the direction opposite to the center, so the distance is radius - distance_between_centers. Wait, maybe.Wait, think of it as follows: imagine a circle centered at C with radius r, and a point P inside the circle. The minimum distance from P to the circle is r - |PC|, and the maximum distance is r + |PC|. Therefore, in this case, since the origin is inside the circle (distance from origin to center is sqrt(2), radius is 2), then the minimum |c| is 2 - sqrt(2), and maximum is 2 + sqrt(2). Therefore, the range is [2 - sqrt(2), 2 + sqrt(2)], which is option B. That seems to match.Wait, but let me confirm this with coordinates. Suppose we have a circle centered at (1,1) with radius 2. The origin is (0,0). The closest point on the circle to the origin would be along the line from the origin through the center (1,1). The direction vector from the origin to the center is (1,1). The unit vector in that direction is (1/√2, 1/√2). The point on the circle in that direction from the center would be (1,1) + 2*(1/√2, 1/√2) = (1 + 2/√2, 1 + 2/√2). The distance from the origin to this point is sqrt[(1 + 2/√2)^2 + (1 + 2/√2)^2] = sqrt[2*(1 + 2/√2)^2] = sqrt[2]*(1 + 2/√2) = sqrt(2) + 2. Wait, but that's the maximum distance. Wait, but if we go in the opposite direction, from the center towards the origin, but since the origin is inside the circle, the closest point would be in the direction from the center towards the origin. Wait, no, the closest point on the circle to the origin would be along the line connecting the origin and the center, but since the origin is inside the circle, the closest point is in the direction from the center towards the origin, subtracted by the radius. Wait, perhaps not. Let's think.Wait, parametric line from origin to center: (0,0) + t*(1,1), t >= 0. The intersection point of this line with the circle centered at (1,1) with radius 2. The equation of the circle is (x - 1)^2 + (y - 1)^2 = 4. The parametric line is x = t, y = t. Plug into the circle equation:(t - 1)^2 + (t - 1)^2 = 42*(t - 1)^2 = 4(t - 1)^2 = 2t - 1 = ±√2t = 1 ± √2Since the line goes from the origin (t=0) through the center (t=1) and beyond, the points of intersection are at t = 1 + √2 and t = 1 - √2. But t = 1 - √2 is approximately 1 - 1.414 ≈ -0.414, which is in the opposite direction from the origin. But since we are looking for points on the circle in all directions, the intersection points are at t = 1 + √2 and t = 1 - √2. However, the point at t = 1 - √2 is in the direction opposite to the center from the origin. Wait, but since the circle is centered at (1,1), the point at t = 1 - √2 would actually be (1 - √2, 1 - √2). The distance from the origin to this point is sqrt[(1 - √2)^2 + (1 - √2)^2] = sqrt[2*(1 - 2√2 + 2)] = sqrt[2*(3 - 2√2)] = sqrt[6 - 4√2]. Wait, but this seems complicated. Alternatively, maybe we can compute the distance from the origin to the two intersection points.At t = 1 + √2, the point is (1 + √2, 1 + √2). The distance from the origin is sqrt[(1 + √2)^2 + (1 + √2)^2] = sqrt[2*(1 + 2√2 + 2)] = sqrt[2*(3 + 2√2)] = sqrt[6 + 4√2]. Similarly, at t = 1 - √2, the point is (1 - √2, 1 - √2). The distance from the origin is sqrt[(1 - √2)^2 + (1 - √2)^2] = sqrt[2*(1 - 2√2 + 2)] = sqrt[2*(3 - 2√2)].Hmm, but how does this relate to the maximum and minimum |c|? The maximum |c| would be the distance from the origin to the farthest point on the circle, which is in the direction away from the origin, so t = 1 + √2, giving sqrt[6 + 4√2]. The minimum |c| is the distance from the origin to the closest point on the circle, which is in the direction towards the origin from the center, so t = 1 - √2, giving sqrt[6 - 4√2]. But these expressions might simplify. Let's check sqrt[6 + 4√2]. Let's see if that's equal to 2 + sqrt(2):(2 + sqrt(2))^2 = 4 + 4*sqrt(2) + 2 = 6 + 4*sqrt(2). Yes! So sqrt[6 + 4√2] = 2 + sqrt(2). Similarly, sqrt[6 - 4√2]. Let's compute (2 - sqrt(2))^2 = 4 - 4*sqrt(2) + 2 = 6 - 4*sqrt(2). Therefore, sqrt[6 - 4√2] = 2 - sqrt(2). Therefore, the minimum |c| is 2 - sqrt(2) and the maximum is 2 + sqrt(2). Therefore, the range is [2 - sqrt(2), 2 + sqrt(2)], which is option B. That seems to check out.Alternatively, using the earlier reasoning, since the origin is inside the circle, the minimum distance is radius - distance_from_origin_to_center, and maximum is radius + distance_from_origin_to_center. Wait, but radius is 2, distance_from_origin_to_center is sqrt(2). So maximum is 2 + sqrt(2), and minimum is 2 - sqrt(2). That matches the result from coordinates. Therefore, the answer is B.But let me verify with another approach. Let's use algebra. Let me consider vectors in general.Given that a and b are orthonormal vectors (unit vectors, perpendicular). Let me write c as a vector in terms of a and b. Let’s denote c = x*a + y*b. Then |c - a - b| = |(x - 1)a + (y - 1)b| = sqrt[(x - 1)^2 + (y - 1)^2] = 2. Therefore, (x - 1)^2 + (y - 1)^2 = 4. We need to find the range of |c| = sqrt(x^2 + y^2).So, the problem reduces to: given that (x - 1)^2 + (y - 1)^2 = 4, find the range of sqrt(x^2 + y^2). This is equivalent to finding the range of the distance from the origin to points (x, y) on the circle centered at (1,1) with radius 2. Which is exactly the same as the geometric problem we considered earlier.Therefore, using the formula for the distance from a point to a circle: if the circle has center (h, k) and radius r, and the point is (p, q), then the minimum distance is |sqrt[(h - p)^2 + (k - q)^2] - r| and maximum distance is sqrt[(h - p)^2 + (k - q)^2] + r. In this case, the point is (0,0), the circle center is (1,1), radius 2. So the distance between the point and the center is sqrt(1 + 1) = sqrt(2). Therefore, the minimum distance is |sqrt(2) - 2| and maximum is sqrt(2) + 2. Since sqrt(2) ≈ 1.414 < 2, |sqrt(2) - 2| = 2 - sqrt(2). Therefore, the range is [2 - sqrt(2), 2 + sqrt(2)], which is option B. So that's the answer.Alternatively, another method is using coordinates. Let me parametrize the circle (x - 1)^2 + (y - 1)^2 = 4. Let me use polar coordinates. Let x = 1 + 2 cos θ, y = 1 + 2 sin θ. Then |c| = sqrt(x^2 + y^2) = sqrt{(1 + 2 cos θ)^2 + (1 + 2 sin θ)^2}. Expanding this:(1 + 4 cos θ + 4 cos²θ) + (1 + 4 sin θ + 4 sin²θ) = 2 + 4 (cos θ + sin θ) + 4 (cos²θ + sin²θ). Since cos²θ + sin²θ = 1, this simplifies to 2 + 4 (cos θ + sin θ) + 4*1 = 6 + 4 (cos θ + sin θ). Therefore, |c| = sqrt(6 + 4 (cos θ + sin θ)).Now, we need to find the range of sqrt(6 + 4 (cos θ + sin θ)). Let’s analyze the expression inside the square root: 6 + 4 (cos θ + sin θ). The term cos θ + sin θ can be rewritten as sqrt(2) sin(θ + 45°), which has a maximum value of sqrt(2) and a minimum of -sqrt(2). Therefore, the expression becomes 6 ± 4 sqrt(2). Therefore, the maximum value of the expression inside the square root is 6 + 4 sqrt(2), and the minimum is 6 - 4 sqrt(2). Therefore, the maximum |c| is sqrt(6 + 4 sqrt(2)) and the minimum is sqrt(6 - 4 sqrt(2)).But as we saw earlier, sqrt(6 + 4 sqrt(2)) = 2 + sqrt(2) and sqrt(6 - 4 sqrt(2)) = 2 - sqrt(2). Let me verify:(2 + sqrt(2))² = 4 + 4 sqrt(2) + 2 = 6 + 4 sqrt(2). Correct.(2 - sqrt(2))² = 4 - 4 sqrt(2) + 2 = 6 - 4 sqrt(2). Correct.Therefore, sqrt(6 + 4 sqrt(2)) = 2 + sqrt(2) and sqrt(6 - 4 sqrt(2)) = 2 - sqrt(2). Therefore, the range of |c| is [2 - sqrt(2), 2 + sqrt(2)], which is option B.Therefore, the answer is B. Let me just check if any of the other options could be possible. Option C is [sqrt(2), 2 sqrt(2)]. But sqrt(2) is approximately 1.414, 2 sqrt(2) is about 2.828. However, 2 - sqrt(2) is about 0.586, which is less than sqrt(2), so the lower bound in B is lower than in C. Similarly, 2 + sqrt(2) is about 3.414, which is higher than 2 sqrt(2). So option C is too narrow. Option D is [3 - 2 sqrt(2), 3 + 2 sqrt(2)]. 3 - 2 sqrt(2) is about 3 - 2.828 = 0.172, and 3 + 2 sqrt(2) is about 5.828. But our range is up to 3.414, so D is too wide. Option A is [1, 1 + sqrt(2)], which is too small. So B is the correct answer.Therefore, after considering multiple approaches—geometric interpretation, coordinate algebra, parametrization—I arrive at the conclusion that the correct answer is B.Final Answerboxed{B}</think>Given that (vec{a}) and (vec{b}) are unit vectors and perpendicular to each other, and vector (vec{c}) satisfies (|vec{c} - vec{a} - vec{b}| = 2), we need to find the range of values for (|vec{c}|).1. Geometric Interpretation: - Since (vec{a}) and (vec{b}) are orthonormal, they form a coordinate system. Let (vec{a} = (1, 0)) and (vec{b} = (0, 1)). - The vector (vec{a} + vec{b} = (1, 1)) has magnitude (sqrt{2}). - The vector (vec{c}) lies on a circle centered at ((1, 1)) with radius 2.2. Distance from Origin: - The distance from the origin ((0, 0)) to the center ((1, 1)) is (sqrt{2}). - Since the origin is inside the circle (radius 2 > (sqrt{2})), the minimum distance from the origin to the circle is (2 - sqrt{2}) and the maximum distance is (2 + sqrt{2}).3. Algebraic Verification: - Parametrize (vec{c}) as ((1 + 2costheta, 1 + 2sintheta)). - Calculate (|vec{c}| = sqrt{(1 + 2costheta)^2 + (1 + 2sintheta)^2}). - Simplify to find (|vec{c}| = sqrt{6 + 4(costheta + sintheta)}). - The expression (costheta + sintheta) ranges from (-sqrt{2}) to (sqrt{2}), leading to the range ([2 - sqrt{2}, 2 + sqrt{2}]).4. Conclusion: - The range of (|vec{c}|) is ([2 - sqrt{2}, 2 + sqrt{2}]), which corresponds to option B.(boxed{B})]

🔑:AnalysisThis problem examines the operation of the scalar product of plane vectors. According to the problem statement, drawing a diagram and combining numerical and geometric methods is a powerful tool for solving this problem. Let overset{→}{OA}= overset{→}{a}, overset{→}{OB}= overset{→}{b}, overset{→}{OD}= overset{→}{a}+ overset{→}{b}, overset{→}{OC}= overset{→}{c}. By drawing the diagram, we can find the maximum and minimum values of |overset{→}{c}|.SolutionLet overset{→}{OA}= overset{→}{a}, overset{→}{OB}= overset{→}{b}, overset{→}{OD}= overset{→}{a}+ overset{→}{b}, overset{→}{OC}= overset{→}{c}.Then, |overset{→}{OD}|= sqrt{2}.Since |overset{→}{c}- overset{→}{a}- overset{→}{b}|=2, point C lies on the circle centered at point D with a radius of 2.It is easy to see that when point C is collinear with O and D, |overset{→}{OC}| reaches its extreme values, with the maximum value being sqrt{2}+2 and the minimum value being 2- sqrt{2}.Therefore, the range of values for |overset{→}{c}| is boxed{[2- sqrt{2} , sqrt{2}+2 ]}.Hence, the correct option is boxed{text{B}}.

🧠:[<think>Okay, let's see. I have this problem here: If 2^m = 3 and 4^n = 8, then what is the value of 2^(3m - 2n + 3)? Hmm. Alright, let me try to figure this out step by step. First, I need to find expressions for m and n, maybe? Because the exponents involve m and n, and I need to compute 2 raised to some combination of them. Let me start with the given equations.So, the first equation is 2^m = 3. That seems straightforward. If I take the logarithm base 2 of both sides, I can solve for m. Similarly, the second equation is 4^n = 8. Hmm, 4 is 2 squared, and 8 is 2 cubed. Maybe I can rewrite the second equation in terms of base 2 to make it easier. Let me try that.Starting with 4^n = 8. Since 4 is 2^2, substituting that in gives (2^2)^n = 8. Which simplifies to 2^(2n) = 2^3. Since the bases are the same, the exponents must be equal. So, 2n = 3, which means n = 3/2. Got that. So n is 3/2.Now, going back to the first equation: 2^m = 3. To solve for m, I can take the logarithm base 2 of both sides. So m = log2(3). Hmm, that's an exact expression, but maybe I can leave it as log2(3) for now since I might not need a numerical value. Let me check.The expression I need to compute is 2^(3m - 2n + 3). Let's break this down. If I can express 3m - 2n + 3 in terms of exponents that I can compute, that might help. But since the base is 2, maybe I can use properties of exponents to split this into separate terms.First, let's substitute the values of m and n into the exponent. So, 3m - 2n + 3. Since m is log2(3), then 3m is 3*log2(3). Similarly, n is 3/2, so 2n is 2*(3/2) which is 3. Therefore, the exponent becomes 3*log2(3) - 3 + 3. Wait, that simplifies to 3*log2(3) - 3 + 3. The -3 and +3 cancel each other out, so it's just 3*log2(3). Therefore, the entire expression 2^(3m - 2n + 3) simplifies to 2^{3*log2(3)}.Hmm, but maybe I made a mistake here. Let me double-check. Let's go step by step. So, 3m - 2n + 3. Substitute m = log2(3) and n = 3/2.3m = 3*log2(3)2n = 2*(3/2) = 3So, 3m - 2n + 3 = 3*log2(3) - 3 + 3. Then, the -3 and +3 do cancel, so yeah, that leaves 3*log2(3). So, the exponent is 3*log2(3), so the expression is 2^{3*log2(3)}.Now, let's recall a logarithmic identity here. The exponent is 3*log2(3), so 2^{a*log2(b)} is equal to b^a. Because in general, c^{d*log_c(e)} = e^d. Let me verify that. Let's take log base c of e, which is log_c(e). If I multiply that by d, I get d*log_c(e). Then c raised to that power is c^{d*log_c(e)}. Using the logarithm power rule, log_c(e^d) = d*log_c(e). Therefore, c^{d*log_c(e)} = e^d. So yes, that identity holds.Therefore, applying that here, 2^{3*log2(3)} = 3^3 = 27. So, the value of 2^{3m - 2n + 3} is 27. Wait, that seems too straightforward. Let me check again.Alternatively, maybe there's another way to approach this problem. Let's try another method to confirm.Given 2^m = 3, so m = log2(3). 4^n = 8, which we converted to n = 3/2. Then, 2^{3m - 2n + 3} can be broken down using exponent rules. Let's split the exponent into separate terms:2^{3m} * 2^{-2n} * 2^3. Because 2^{a + b + c} = 2^a * 2^b * 2^c.So, 2^{3m} is (2^m)^3. Since 2^m = 3, that would be 3^3 = 27.Then, 2^{-2n} is 1/(2^{2n}). But 2^{2n} can be written as (2^2)^n = 4^n. Given that 4^n = 8, so 2^{2n} = 8, so 2^{-2n} = 1/8.Then, 2^3 is 8.Putting it all together: 27 * (1/8) * 8. The 1/8 and 8 cancel each other out, leaving 27. So, that's another way to get 27. So that seems to confirm the answer is 27.Wait, so both methods give me 27, so that must be correct. Let me just check again each step in case I missed something.First, solving for n: 4^n = 8. Since 4 is 2^2, that becomes (2^2)^n = 2^3. Then, 2^{2n} = 2^3. Therefore, 2n = 3, so n = 3/2. That seems right.Then, solving for m: 2^m = 3. So m = log2(3). That's correct.Then, substituting into 2^{3m - 2n + 3}. Let's write that exponent as 3m - 2n + 3. Substituting m = log2(3) and n = 3/2:3m = 3*log2(3)2n = 2*(3/2) = 3So, exponent is 3*log2(3) - 3 + 3 = 3*log2(3). Then 2^{3*log2(3)} = 3^3 = 27. Yes.Alternatively, splitting into 2^{3m} * 2^{-2n} * 2^3. Then:2^{3m} = (2^m)^3 = 3^3 = 272^{-2n} = (2^{2n})^{-1} = (4^n)^{-1} = 8^{-1} = 1/82^3 = 8Multiply all: 27 * (1/8) * 8 = 27. Same result.Therefore, regardless of the method, the answer is 27. So I think that's solid.Wait, but let me just think if there's another perspective. Maybe using natural logarithms? Let's see. If I take ln on both sides of the original equations.For 2^m = 3, taking ln: m*ln2 = ln3 => m = ln3 / ln2Similarly, 4^n = 8 => ln(4^n) = ln8 => n*ln4 = ln8 => n = ln8 / ln4Then, compute 3m - 2n + 3:3*(ln3 / ln2) - 2*(ln8 / ln4) + 3But ln4 is 2ln2, and ln8 is 3ln2. Therefore,3*(ln3 / ln2) - 2*(3ln2 / 2ln2) + 3Simplify the second term: 2*(3ln2 / 2ln2) = 3So, 3*(ln3 / ln2) - 3 + 3 = 3*(ln3 / ln2)Therefore, exponent is 3*(ln3 / ln2). Then 2^{3*(ln3 / ln2)}.But 2^{ln3 / ln2} is equal to e^{(ln3 / ln2)*ln2} = e^{ln3} = 3. So, 2^{ln3 / ln2} = 3. Therefore, 2^{3*(ln3 / ln2)} = 3^3 = 27. Same result.So, even using natural logarithms, we still end up with 27. Therefore, this seems to confirm the answer again. So, yeah, 27 is the answer.But just to make sure, let's try plugging in approximate numbers. Let me compute m and n numerically.Given 2^m = 3. Taking log base 2, m ≈ 1.58496 (since 2^1.58496 ≈ 3). Similarly, n = 3/2 = 1.5.Then, the exponent is 3m - 2n + 3. Let's compute that:3*1.58496 ≈ 4.754882*1.5 = 3So, 4.75488 - 3 + 3 = 4.75488Therefore, 2^4.75488 ≈ ?Since 2^4 = 16, 2^5 = 32. 4.75488 is between 4 and 5. Let's compute 2^4.75488.But wait, 3^3 = 27. So, if exponent is log2(27), which is log2(3^3) = 3*log2(3) ≈ 3*1.58496 ≈ 4.75488. So, 2^4.75488 = 27. So, yes, numerically, it checks out. Therefore, that's consistent.Therefore, all methods lead to the same answer, 27. So, I think that's the correct answer. It makes sense. The problem was designed to cancel out some terms, maybe. So, yeah, 27.Final AnswerThe value of (2^{3m - 2n + 3}) is boxed{27}.</think>Given the equations (2^m = 3) and (4^n = 8), we need to find the value of (2^{3m - 2n + 3}).First, solve for (m) and (n):1. From (2^m = 3), we take the logarithm base 2 of both sides: [ m = log_2 3 ]2. From (4^n = 8), we rewrite 4 as (2^2) and 8 as (2^3): [ (2^2)^n = 2^3 implies 2^{2n} = 2^3 implies 2n = 3 implies n = frac{3}{2} ]Next, substitute (m) and (n) into the exponent (3m - 2n + 3):1. Calculate (3m): [ 3m = 3 log_2 3 ]2. Calculate (2n): [ 2n = 2 left(frac{3}{2}right) = 3 ]3. Combine the terms: [ 3m - 2n + 3 = 3 log_2 3 - 3 + 3 = 3 log_2 3 ]The expression (2^{3m - 2n + 3}) simplifies to (2^{3 log_2 3}). Using the property of exponents (2^{a log_2 b} = b^a):[2^{3 log_2 3} = 3^3 = 27]Thus, the value of (2^{3m - 2n + 3}) is (boxed{27}).,<think>Okay, so I need to find the value of (2^{3m - 2n + 3}) given that (2^m = 3) and (4^n = 8). Hmm, let's start by writing down what we know. First, the problem gives two equations: (2^m = 3) and (4^n = 8). The question is asking for the value of an exponent expression with base 2, so maybe I can express everything in terms of base 2 to make it easier. Let me handle each equation one by one. The first one is straightforward: (2^m = 3). That means if I need (m), I can take the logarithm base 2 of both sides. But maybe I don't need the actual value of (m) and (n) but instead can manipulate the exponents directly. Let's see.The second equation is (4^n = 8). Since 4 and 8 are both powers of 2, I can rewrite them as such. So 4 is (2^2) and 8 is (2^3). Therefore, substituting into the second equation, we get ((2^2)^n = 2^3). Simplifying the left side, when you raise a power to a power, you multiply the exponents. So that's (2^{2n} = 2^3). Since the bases are the same and the equation holds, the exponents must be equal. Therefore, (2n = 3), so (n = frac{3}{2}). Okay, so now I know that (n = frac{3}{2}). Let me note that down. Now, going back to the first equation, (2^m = 3). Again, if I need (m), I can take the logarithm base 2 of both sides. So (m = log_2 3). But similar to before, maybe I don't need the actual value of (m), but instead express other exponents in terms of (m). Let's see what the question is asking for.The question wants (2^{3m - 2n + 3}). Hmm, exponent rules tell me that this can be broken down into separate factors. Specifically, (2^{3m - 2n + 3} = 2^{3m} times 2^{-2n} times 2^3). Alternatively, since addition in the exponent is multiplication of the bases, and subtraction is division. Let me write that out:(2^{3m - 2n + 3} = 2^{3m} times 2^{-2n} times 2^3).Which can also be written as:(2^{3m} times frac{1}{2^{2n}} times 8). Because (2^3 = 8). Now, if I can express (2^{3m}) and (2^{2n}) in terms of the given equations, maybe I can substitute the known values.Starting with (2^{3m}): since (2^m = 3), raising both sides of this equation to the power of 3 gives ((2^m)^3 = 3^3), which is (2^{3m} = 27). So that's straightforward. Next, (2^{2n}): from the second equation, we found that (4^n = 8), which we rewrote as (2^{2n} = 8). Therefore, (2^{2n} = 8), so (2^{2n} = 8). Therefore, (2^{2n} = 8), which is already given. So substituting back into the expression for (2^{3m - 2n + 3}):(2^{3m} times frac{1}{2^{2n}} times 8 = 27 times frac{1}{8} times 8).Wait, let's see. (2^{3m}) is 27, (2^{2n}) is 8, so (1/2^{2n}) is (1/8), and then multiplied by 8. So that would be 27 * (1/8) * 8. The 1/8 and 8 cancel each other out, so we just get 27.Wait, is that right? Let me check again step by step.Starting with:(2^{3m - 2n + 3} = 2^{3m} times 2^{-2n} times 2^3).We have:- (2^{3m}): From (2^m = 3), so ( (2^m)^3 = 3^3 implies 2^{3m} = 27 ).- (2^{-2n}): Since (2^{2n} = 8), then (2^{-2n} = 1/8).- (2^3 = 8).Putting them all together: 27 * (1/8) * 8 = 27 * (8/8) = 27 * 1 = 27.So the answer is 27. Wait, that seems straightforward, but let me verify again to make sure I didn't miss anything.Alternatively, maybe I can compute (3m - 2n + 3) first and then compute 2 raised to that exponent. Let's try that approach.Given that (2^m = 3) and (4^n = 8), let's find m and n.From (2^m = 3), taking logarithm base 2: (m = log_2 3).From (4^n = 8), as we did before, (4^n = 8 implies (2^2)^n = 2^3 implies 2^{2n} = 2^3 implies 2n = 3 implies n = 3/2).So m is (log_2 3) and n is (3/2). Then compute (3m - 2n + 3):3m = 3 * (log_2 3),2n = 2 * (3/2) = 3,So 3m - 2n + 3 = 3(log_2 3) - 3 + 3 = 3(log_2 3).Therefore, (2^{3m - 2n + 3} = 2^{3 log_2 3}).But (2^{log_2 3} = 3), so (2^{3 log_2 3} = (2^{log_2 3})^3 = 3^3 = 27). So both methods give me 27. That's reassuring. Wait, so in the second approach, I initially thought that 3m -2n +3 equals 3 log_2 3, but let me check again:3m -2n +3 = 3 log_2 3 - 2*(3/2) +3= 3 log_2 3 - 3 +3= 3 log_2 3Yes, that's correct. Because 2n is 2*(3/2) = 3, then subtract 3 and add 3, which cancels out. So the exponent simplifies to 3 log_2 3, which is the same as log_2 3^3 = log_2 27. Therefore, 2^{log_2 27} = 27. Either way, both methods confirm that the answer is 27. Therefore, the value of (2^{3m - 2n +3}) is 27.Final Answerboxed{27}</think>Given the equations (2^m = 3) and (4^n = 8), we need to find the value of (2^{3m - 2n + 3}).First, we rewrite the second equation in terms of base 2:[4^n = 8 implies (2^2)^n = 2^3 implies 2^{2n} = 2^3 implies 2n = 3 implies n = frac{3}{2}]Next, we use the first equation (2^m = 3). We need to express (2^{3m}) and (2^{2n}):[2^{3m} = (2^m)^3 = 3^3 = 27][2^{2n} = 8 implies 2^{-2n} = frac{1}{8}]Now, we substitute these values into the expression (2^{3m - 2n + 3}):[2^{3m - 2n + 3} = 2^{3m} times 2^{-2n} times 2^3 = 27 times frac{1}{8} times 8]Simplifying the expression:[27 times frac{1}{8} times 8 = 27 times 1 = 27]Thus, the value of (2^{3m - 2n + 3}) is (boxed{27}).]

🔑:Given 2^{m}=3 and 4^{n}=8, we have 2^{3m-2n+3}=(2^{m})^{3}÷(2^{n})^{2}×2^{3}, which simplifies to (2^{m})^{3}÷4^{n}×2^{3}, and further to 3^{3}÷8×8, resulting in 27. Therefore, the answer is boxed{27}. This problem involves the application of the properties of exponents, specifically the division of powers with the same base and the property of raising a power to another power, in reverse. The key to solving the problem is to express 2^{3m-2n+3} as (2^{m})^{3}÷(2^{n})^{2}×2^{3}, and then substitute the given values. This showcases the importance of both directly applying and inversely applying the rules of exponents in solving algebraic expressions, a crucial skill in middle school mathematics competitions.

🧠:[<think>Okay, so I need to find all values of α where the curves y = αx² + αx + 1/24 and x = αy² + αy + 1/24 are tangent to each other. Hmm, tangency means they touch at a single point and share the same slope there. Right? So, first, I should find the points where these two curves intersect and then set their derivatives equal at those points. Then solve for α. Let me break this down step by step.First, let's consider the equations:1. y = αx² + αx + 1/242. x = αy² + αy + 1/24Since they are tangent, they must intersect at exactly one point (or maybe multiple points but with at least one point where their slopes are equal). Wait, but tangency usually means they touch at one point and don't cross each other there. So maybe there's only one point of intersection where the slopes are equal. Or perhaps multiple points but at least one with equal slope. The problem says "tangent to each other," so I think it's sufficient that there exists at least one point where they are tangent, regardless of other intersection points. Hmm, maybe I should check both possibilities.But to start, let's suppose that the curves are tangent at some point (a, b). Then (a, b) lies on both curves, so substituting into both equations:b = αa² + αa + 1/24 ... (1)a = αb² + αb + 1/24 ... (2)Additionally, the derivatives at that point should be equal. Let's compute the derivatives.For the first curve, dy/dx = 2αx + α.For the second curve, since x is expressed in terms of y, we can differentiate both sides with respect to x:dx/dx = d/dx [αy² + αy + 1/24]1 = α*2y*dy/dx + α*dy/dx + 0So, 1 = α(2y + 1) dy/dxTherefore, dy/dx = 1 / [α(2y + 1)] ... (assuming α ≠ 0 and 2y + 1 ≠ 0)Therefore, at the point of tangency (a, b), the derivatives must be equal:2αa + α = 1 / [α(2b + 1)]So, (2a + 1)α = 1 / [α(2b + 1)]Multiply both sides by α(2b + 1):(2a + 1)α²(2b + 1) = 1 ... (3)So now we have three equations: (1), (2), and (3). We need to solve these equations for a, b, and α.But this seems complicated. Let me see if I can find a relationship between a and b.From equation (1): b = αa² + αa + 1/24From equation (2): a = αb² + αb + 1/24These are symmetric equations. Maybe a = b? Let me check if that's possible.Suppose a = b. Then substituting into equation (1):a = αa² + αa + 1/24Bring all terms to one side:αa² + αa + 1/24 - a = 0αa² + (α - 1)a + 1/24 = 0 ... (4)Similarly, substituting a = b into equation (2) would give the same equation. So if a = b is possible, then equation (4) must hold. Additionally, the derivative condition (equation (3)) would become:(2a + 1)α²(2a + 1) = 1So, (2a + 1)^2 α² = 1 ... (5)Therefore, if we can find a such that equation (4) and (5) hold, then a = b is a possible point of tangency.So let's proceed under the assumption that a = b. Then we have equations (4) and (5):From equation (4):αa² + (α - 1)a + 1/24 = 0From equation (5):(2a + 1)^2 α² = 1We can solve equation (5) for α:Take square roots:(2a + 1)α = ±1Thus, α = ±1/(2a + 1) ... (6)Now, substitute α from equation (6) into equation (4):Let's take α = 1/(2a + 1) first.Substituting into equation (4):[1/(2a + 1)]a² + [1/(2a + 1) - 1]a + 1/24 = 0Multiply through by (2a + 1) to eliminate denominators:a² + [1 - (2a + 1)]a + (1/24)(2a + 1) = 0Simplify:a² + [1 - 2a -1]a + (2a +1)/24 = 0a² + (-2a²) + (2a +1)/24 = 0Combine like terms:- a² + (2a +1)/24 = 0Multiply through by 24 to eliminate the denominator:-24a² + 2a +1 = 0Multiply by -1:24a² -2a -1 = 0Solve this quadratic equation:a = [2 ± sqrt(4 + 96)] / 48sqrt(100) = 10So,a = [2 ± 10]/48Thus, a = (12)/48 = 1/4 or a = (-8)/48 = -1/6So, possible a values are 1/4 and -1/6. Let's find α for each.For a = 1/4:From equation (6), α = 1/(2*(1/4) +1) = 1/(1/2 +1) = 1/(3/2) = 2/3For a = -1/6:α = 1/(2*(-1/6) +1) = 1/(-1/3 +1) = 1/(2/3) = 3/2Now check the other case where α = -1/(2a +1)Substitute α = -1/(2a +1) into equation (4):[-1/(2a +1)]a² + [-1/(2a +1) -1]a +1/24 =0Multiply through by (2a +1):- a² + [-1 - (2a +1)]a + (2a +1)/24 =0Simplify:- a² + [-1 -2a -1]a + (2a +1)/24 =0- a² + (-2a -2)a + (2a +1)/24 =0- a² -2a² -2a + (2a +1)/24 =0Combine terms:-3a² -2a + (2a +1)/24 =0Multiply through by 24:-72a² -48a +2a +1 =0Simplify:-72a² -46a +1 =0Multiply by -1:72a² +46a -1 =0Solve this quadratic:a = [-46 ± sqrt(46² +4*72*1)]/(2*72)Calculate discriminant:46² = 21164*72*1 = 288Total discriminant: 2116 +288 = 2404sqrt(2404). Let's check sqrt(2404). 49²=2401, so sqrt(2404)=sqrt(2401+3)=49.04... Hmm, but exact value? Let's see:2404 divided by 4 is 601, which is prime? 601 ÷ 7=85.85..., not integer. So sqrt(2404) = 2*sqrt(601). So,a = [-46 ± 2√601]/144 = [-23 ± √601]/72Hmm, messy. Let's check if these a's lead to real solutions. But maybe this case (α negative) is possible. But let's check if these a's result in α being real.From α = -1/(2a +1), so denominator 2a +1.Compute 2a +1 for a = (-23 +√601)/72:2a +1 = 2*(-23 +√601)/72 +1 = (-46 + 2√601)/72 + 72/72 = (26 + 2√601)/72 = (13 + √601)/36 ≠0Similarly for the other root a = (-23 -√601)/72, denominator is (13 -√601)/36. Since √601≈24.5, 13 -24.5≈-11.5, so denominator is negative, but not zero. So α is defined.But these solutions seem complicated, and maybe these are extraneous. Let's check if these a and α actually satisfy the original equations.But before that, let's check the first case where α was positive. We found α=2/3 and α=3/2. Let me check if these work.First, α=2/3. Let's verify if the curves are tangent.Substitute α=2/3 into both equations:First curve: y = (2/3)x² + (2/3)x +1/24Second curve: x = (2/3)y² + (2/3)y +1/24If a=1/4, then b=1/4. Let's check if (1/4,1/4) lies on both curves.First curve: y = (2/3)(1/4)^2 + (2/3)(1/4) +1/24Compute:(2/3)(1/16) + (2/3)(1/4) +1/24 = (2/48) + (8/48) + 2/48 = (2 +8 +2)/48 =12/48=1/4. Correct.Second curve: x = (2/3)(1/4)^2 + (2/3)(1/4) +1/24. Same computation, which also gives 1/4. Correct.Now check the derivatives at (1/4,1/4):For first curve: dy/dx = 2*(2/3)*x + (2/3). At x=1/4:= (4/3)(1/4) + 2/3 = 1/3 + 2/3 =1For second curve: dy/dx = 1 / [α(2y +1)] = 1 / [(2/3)(2*(1/4)+1)] =1 / [(2/3)(1/2 +1)] =1 / [(2/3)(3/2)] =1 /1=1. So derivatives are equal. So yes, α=2/3 is valid.Similarly, check α=3/2. Then a=-1/6, so b=-1/6.Check if (-1/6, -1/6) lies on both curves.First curve: y = (3/2)x² + (3/2)x +1/24Compute y:(3/2)(1/36) + (3/2)(-1/6) +1/24 = (3/72) - (3/12) +1/24 = (1/24) - (1/4) +1/24 = (1/24 +1/24) -1/4 = (2/24) -6/24 = -4/24 = -1/6. Correct.Second curve: x = (3/2)y² + (3/2)y +1/24Similarly:(3/2)(1/36) + (3/2)(-1/6) +1/24 = same as above, -1/6. Correct.Derivatives:First curve: dy/dx = 2*(3/2)x + (3/2) = 3x + 3/2. At x=-1/6:3*(-1/6) +3/2 = -1/2 +3/2=1.Second curve: dy/dx=1/[α(2y +1)] =1/[ (3/2)(2*(-1/6)+1) ]=1/[ (3/2)(-1/3 +1) ]=1/[ (3/2)(2/3) ]=1/(1)=1. Equal derivatives. So α=3/2 is also valid.So these are two valid α's: 2/3 and 3/2.Now, what about the other case where α is negative? Let's check with a = [ -23 + sqrt(601) ] /72 and α = -1/(2a +1). Let me compute numerical values to check.Compute sqrt(601) ≈24.51, so a ≈ (-23 +24.51)/72 ≈1.51/72≈0.021. Then 2a +1≈1.042, so α≈-1/1.042≈-0.96.Similarly, other root a≈(-23 -24.51)/72≈-47.51/72≈-0.66. Then 2a +1≈2*(-0.66)+1≈-1.32+1≈-0.32, so α≈-1/(-0.32)=3.125.But maybe these α's also give valid solutions. Let me check if these points satisfy the equations.Take α≈-0.96, a≈0.021. Compute b from equation (1):b=αa² + αa +1/24≈-0.96*(0.021)^2 + (-0.96)(0.021) +1/24≈-0.96*(0.000441) -0.02016 +0.0417≈≈-0.000424 -0.02016 +0.0417≈0.0211. So b≈0.021≈a. Then check equation (2): x=αy² + αy +1/24≈-0.96*(0.021)^2 -0.96*(0.021) +0.0417≈≈same as above≈0.0211≈a. So close. But this is approximate, so maybe exact?But since these come from solving the equations, they should be exact. However, these solutions are messy. Let's see:From equation (4) with α = -1/(2a +1):We had 72a² +46a -1=0. So solutions a=(-23 ±√(23² +72))/72= (-23 ±√(529 +72))/72= (-23 ±√601)/72.Then α= -1/(2a +1). Let's compute α for a= [ -23 +√601 ] /72.Compute 2a +1= 2*(-23 +√601)/72 +1= (-46 + 2√601)/72 +72/72= (26 + 2√601)/72= (13 +√601)/36.Thus α= -1 / [ (13 +√601)/36 ] = -36/(13 +√601). Rationalize denominator:Multiply numerator and denominator by (13 -√601):α= -36*(13 -√601)/[ (13)^2 - (√601)^2 ]= -36*(13 -√601)/(169 -601)= -36*(13 -√601)/(-432)= (36*(13 -√601))/432= (13 -√601)/12≈(13 -24.51)/12≈-11.51/12≈-0.959.Similarly for a= [ -23 -√601 ] /72:2a +1= 2*(-23 -√601)/72 +1= (-46 -2√601)/72 +72/72= (26 -2√601)/72= (13 -√601)/36.Thus α= -1 / [ (13 -√601)/36 ]= -36/(13 -√601). Again, rationalize:Multiply numerator and denominator by (13 +√601):α= -36*(13 +√601)/[169 -601]= -36*(13 +√601)/(-432)= (36*(13 +√601))/432= (13 +√601)/12≈(13 +24.51)/12≈37.51/12≈3.125.So α≈-0.959 and α≈3.125. Wait, but we already have α=3/2=1.5 and α=2/3≈0.666. So these new α's are different. Let's check if these indeed result in the curves being tangent.But let's test α=(13 -√601)/12≈-0.96. Let's take a point (a, b) where a≈0.021. Then b≈0.021. Let's compute the derivatives.First curve: dy/dx=2αx +α≈2*(-0.96)(0.021) + (-0.96)≈-0.04032 -0.96≈-1.00032.Second curve: dy/dx=1/[α(2y +1)]≈1/[(-0.96)(2*0.021 +1)]≈1/[(-0.96)(1.042)]≈1/(-1.00032)≈-0.999≈-1.000. So derivatives are approximately equal. So close. But due to rounding, but exact computation would show equality.But let's do it symbolically. Let me take α=(13 -√601)/12. Then 2a +1=(13 +√601)/36. Wait, no, earlier we had:From equation (5): (2a +1)^2 α² =1. If α=(13 -√601)/12, then let's compute α²:[(13 -√601)/12]^2 = [169 -26√601 +601]/144 = [770 -26√601]/144.(2a +1)^2= [ (13 +√601)/36 ]^2 = [169 +26√601 +601]/1296 = [770 +26√601]/1296.Then (2a +1)^2 α²= [770 +26√601]/1296 * [770 -26√601]/144= [770² - (26√601)^2 ]/(1296*144).Compute numerator: 770² - (26^2)(601).770²=592,900.26²=676.676*601=676*(600+1)=405,600 +676=406,276.Thus, numerator=592,900 -406,276=186,624.Denominator=1296*144= (1296=36², 144=12²), so 36²*12²= (36*12)^2=432²=186,624.Thus, (2a +1)^2 α²=186,624 /186,624=1. So equation (5) is satisfied.Therefore, these solutions are valid.But wait, we already have α=(13 -√601)/12≈-0.96 and α=(13 +√601)/12≈(13 +24.5)/12≈37.5/12≈3.125. So these are two more solutions.But wait, earlier we found α=2/3≈0.666 and α=3/2=1.5. So total four solutions?But hold on, when we assumed a = b, we found two solutions. When we considered a ≠b, do we get more solutions? Wait, but in our initial approach, we assumed a = b. If we don't assume that, then we have to solve the equations:From (1): b = αa² + αa +1/24From (2): a = αb² + αb +1/24From (3): (2a +1)α²(2b +1) =1So three equations with variables a, b, α. This system is symmetric in a and b only if a = b. If a ≠b, then we have to solve asymmetric equations. But due to the symmetry of the original equations, maybe non-symmetric solutions come in pairs? Not sure.But perhaps all solutions must have a = b? Let's check.Suppose there is a solution where a ≠b. Then due to the symmetry of the equations, swapping a and b would also give a solution. So for each solution (a, b), (b, a) is also a solution. But tangency requires only one point? Wait, but if there are two distinct points of intersection, each with their own slopes, tangency could occur at one of them even if another exists. But the problem says "the curves are tangent to each other," which might mean that they are tangent (at least at one point), regardless of other intersections. So maybe even if there are two points of intersection, as long as at least one of them is a point of tangency, then α is valid.But how to check if such solutions exist? Alternatively, maybe the only possible points of tangency are those where a = b. Because otherwise, solving the three equations is complicated. Let's try to see.Suppose a ≠b.From (1): b = αa² + αa +1/24From (2): a = αb² + αb +1/24Let me subtract these two equations:b - a = α(a² +a - b² -b)Factor the RHS:α[(a² - b²) + (a - b)] = α(a - b)(a + b +1)Thus,b -a = α(a - b)(a + b +1)Multiply both sides by -1:a - b = α(a - b)(a + b +1)If a ≠b, we can divide both sides by (a - b):1 = α(a + b +1)So, we get:α(a + b +1) =1 ... (7)So if a ≠b, then equation (7) must hold. So in addition to equations (1) and (2), and (3), we have equation (7).Additionally, equation (3): (2a +1)(2b +1)α² =1So, now we have four equations: (1), (2), (3), (7). Wait, but (7) is a result of (1) and (2) when a ≠b, so actually three equations.Let me write them again:1. b = αa² + αa +1/242. a = αb² + αb +1/243. (2a +1)(2b +1)α² =14. α(a + b +1) =1So let's use equation (4): α =1/(a + b +1)Substitute this into equations (1) and (2):From (1):b = [1/(a + b +1)]a² + [1/(a + b +1)]a +1/24Multiply both sides by (a + b +1):b(a + b +1) =a² +a + (a + b +1)/24Similarly, from (2):a(a + b +1) =b² +b + (a + b +1)/24These look complex, but let's denote S =a + b and P =ab. Maybe express in terms of S and P.But maybe subtract the two equations.From equation (1) multiplied out:b(a + b +1) =a² +a + (a + b +1)/24Similarly, equation (2):a(a + b +1) =b² +b + (a + b +1)/24Subtract equation (2) from equation (1):b(a + b +1) -a(a + b +1) =a² +a -b² -bLeft side: (b -a)(a + b +1)Right side: (a² - b²) + (a -b) = (a -b)(a + b) + (a -b) = (a -b)(a + b +1)Thus, (b -a)(a + b +1) = (a -b)(a + b +1)Which simplifies to:-(a -b)(a + b +1) = (a -b)(a + b +1)If a ≠b, then (a -b) ≠0, so divide both sides:-(a + b +1) = (a + b +1)Thus, - (a + b +1) = a + b +1 => -a -b -1 =a +b +1 => -2a -2b -2 =0 => a + b +1 =0.But from equation (7): α(a + b +1)=1. If a + b +1=0, then α*0=1, which is impossible. Therefore, contradiction. Therefore, our assumption that a ≠b leads to a contradiction unless a + b +1=0, but that would make equation (7) impossible.Therefore, the only possible solutions are when a =b. Hence, all points of tangency must occur where a =b.Therefore, the only possible solutions are the ones we found earlier with a =b, leading to α=2/3 and α=3/2, as well as the other two solutions from the negative α case. Wait, but we had two more α's from the negative case. But according to this reasoning, since when a ≠b we reach a contradiction, so those other solutions must actually have a =b. Wait, but we derived equations (4) and (5) under the assumption that a =b. Then when we considered α negative, we found two more solutions, but they still have a =b. Wait, but in those cases, a and b are equal? Wait, no. Wait, when we considered α negative, we still assumed a =b. Let me check.Wait, in the previous steps, when we considered the case where a =b, we found α=2/3 and 3/2 from positive roots, and also considered α negative, leading to other solutions. But according to the above conclusion, even for those negative α's, a =b. Wait, but let's check.For example, take α=(13 -√601)/12≈-0.96, then a= [ -23 +√601 ] /72≈0.021, which would imply b= same as a, since we derived under a =b. Wait, no, in equation (4) and (5), we assumed a =b. Therefore, even for those negative α's, a =b.Therefore, all solutions must have a =b, and hence the only possible α's are the four roots? Wait, no, earlier when we considered a =b, we found two quadratics: one for positive α leading to two solutions, and one for negative α leading to two solutions. But when we checked the negative α case, even though a and b are equal, the computed α's resulted in valid tangency points. But we also saw that when we assumed a ≠b, it's impossible.Wait, but according to the previous reasoning, all solutions must have a =b. Therefore, the four α's are potential solutions. Wait, but earlier when we solved the case for a =b, we had two quadratics, leading to four solutions: two from positive α and two from negative α. Wait, no. Wait, actually, when we assumed a =b, we derived equation (4):αa² + (α -1)a +1/24 =0and equation (5):(2a +1)^2 α²=1From equation (5), we solved α=±1/(2a +1). Then substituted into equation (4).For α=1/(2a +1), we obtained quadratic equation leading to a=1/4 and a=-1/6, giving α=2/3 and 3/2.For α=-1/(2a +1), we obtained another quadratic leading to a≈0.021 and a≈-0.66, giving α≈-0.96 and 3.125. Wait, but according to the previous contradiction, when we assume a ≠b, we end up with no solution, meaning that these four α's (2/3, 3/2, -0.96, 3.125) all correspond to a =b.But wait, the problem is asking for real values of α where the curves are tangent. So perhaps all four are valid? But in reality, when I tested α≈-0.96, the point (a,b)=(0.021,0.021) is on both curves and derivatives are equal. So even though α is negative, the curves are still tangent at that point.But let's check with α=(13 -√601)/12≈-0.96. Let's compute the curves:First curve: y = αx² + αx +1/24≈-0.96x² -0.96x +0.0417Second curve: x = αy² + αy +1/24≈-0.96y² -0.96y +0.0417At the point (a,b)=(0.021,0.021), substituting into first curve:y≈-0.96*(0.00045) -0.96*0.021 +0.0417≈≈-0.00043 -0.02016 +0.0417≈0.0211. Close to 0.021.Similarly for x. So, approximately, it works. So tangency occurs here. So even though α is negative, there is a point of tangency.But why did the previous reasoning say all solutions must have a =b? Because when we assumed a ≠b, we reached a contradiction. So the conclusion is that all solutions must have a =b, so even these negative α's correspond to a =b.Therefore, total solutions are α=2/3, 3/2, (13 -√601)/12, and (13 +√601)/12. Wait, but in our earlier calculation, we had α=(13 -√601)/12≈-0.96 and α=(13 +√601)/12≈3.125. Therefore, four solutions? But the problem says "Find all values of α". But maybe some of these are extraneous? Let's check.Wait, when we derived the equations under a =b, we found two quadratics. For the case α=1/(2a +1), leading to two real solutions. For the case α=-1/(2a +1), leading to two real solutions. So total four solutions. But when we computed numerically, two of them are approximately -0.96 and 3.125, which are distinct from 2/3 and 3/2. Therefore, there are four solutions? But the original problem might expect two solutions. Wait, maybe there's a mistake in assuming that a =b is the only possibility. Wait, but according to the previous analysis, assuming a ≠b leads to a contradiction, so only a =b is possible. Hence, all solutions must have a =b. Therefore, the four α's are indeed solutions. However, when we check the curves for these α's, maybe some of them result in the curves intersecting at more than one point, but being tangent at one of them. But tangency requires that at the point of contact, the curves have the same slope, but they may intersect elsewhere. The problem says "the curves are tangent to each other," which I think means that they have at least one point of tangency, regardless of other intersections. So even if they cross at another point, as long as they are tangent at one, α is valid.But let's check for α=3.125 (approx 3.125). Let's see:First curve: y=3.125x² +3.125x +1/24≈3.125x² +3.125x +0.0417Second curve: x=3.125y² +3.125y +0.0417Suppose a≈-0.66, so (a,b)=(-0.66, -0.66). Let's check:First curve: y≈3.125*(0.4356) +3.125*(-0.66) +0.0417≈1.36125 -2.0625 +0.0417≈≈-0.66. Correct.Second curve: x≈3.125*(0.4356) +3.125*(-0.66) +0.0417≈same≈-0.66. Correct.Derivatives:First curve: dy/dx=2*3.125x +3.125. At x=-0.66≈2*3.125*(-0.66)+3.125≈≈-4.125 +3.125≈-1.Second curve: dy/dx=1/[α(2y +1)]≈1/[3.125*(2*(-0.66)+1)]≈1/[3.125*(-1.32 +1)]≈1/[3.125*(-0.32)]≈1/[-1.0]≈-1. So derivatives equal. Thus, tangency at (-0.66, -0.66). However, let's see if there are other intersection points.Suppose we set y=3.125x² +3.125x +0.0417 and x=3.125y² +3.125y +0.0417. If we substitute y from the first equation into the second:x=3.125*(3.125x² +3.125x +0.0417)^2 +3.125*(3.125x² +3.125x +0.0417) +0.0417This would be a quartic equation, which might have more than one real root. So even though they are tangent at (-0.66, -0.66), they might intersect elsewhere. But the problem only requires that they are tangent (at least at one point), so α=3.125 is still valid.Similarly, for α≈-0.96, even if there's another intersection, as long as they are tangent at one point, it's valid.But wait, perhaps for these α's, the curves are tangent but also intersect at another point. Then, is the condition "tangent to each other" satisfied? Yes, because tangency doesn't require that they only touch at one point, just that they are tangent at least at one. So even if they cross at another point, as long as they are tangent somewhere, it's okay.Therefore, in total, there are four solutions for α: 2/3, 3/2, (13 -√601)/12, and (13 +√601)/12.But let's compute (13 -√601)/12 and (13 +√601)/12 exactly.Note that √601 is irrational, so these are two irrational numbers. Alternatively, we can rationalize them, but they can't be simplified further.But maybe we made a mistake here. Wait, when we considered the negative α case, we derived 72a² +46a -1=0. Solving quadratic gives a=(-23 ±√601)/72. Then α= -1/(2a +1). Plugging in:For a=(-23 +√601)/72:α= -1/(2*(-23 +√601)/72 +1)= -1/( (-46 +2√601 +72)/72 )= -1/( (26 +2√601)/72 )= -72/(26 +2√601 )= -72/(2(13 +√601 ))= -36/(13 +√601 )Multiply numerator and denominator by (13 -√601 ):α= -36*(13 -√601 )/(169 -601 )= -36*(13 -√601 )/(-432)= (36*(13 -√601 ))/432= (13 -√601 )/12≈(13 -24.51)/12≈-0.96.Similarly, for a=(-23 -√601)/72:α= -1/(2*(-23 -√601)/72 +1)= -1/( (-46 -2√601 +72)/72 )= -1/(26 -2√601)/72= -72/(26 -2√601 )= -72/(2(13 -√601 ))= -36/(13 -√601 )Multiply numerator and denominator by (13 +√601 ):α= -36*(13 +√601 )/(169 -601 )= -36*(13 +√601 )/(-432)= (36*(13 +√601 ))/432= (13 +√601 )/12≈3.125.Therefore, the exact values are α=(13 ±√601)/12 and α=2/3, 3/2.Wait, but wait, when we solved equation (4) for α=1/(2a +1), we found a=1/4 and a=-1/6, leading to α=2/3 and 3/2. Then for α=-1/(2a +1), we found α=(13 -√601)/12 and (13 +√601)/12. So indeed four solutions. But the problem says "Find all values of α", so we need to present all four? But when I checked with the approximate values, the curves with α≈-0.96 and α≈3.125 also result in tangency. So maybe the answer is all four α's.But wait, in the original problem statement, are we to consider only real numbers? Yes, since tangency in real plane. So, are all four solutions real? Yes.But let me check for α=(13 -√601)/12≈-0.96:Curves:1. y = αx² + αx +1/242. x = αy² + αy +1/24At the point (a,a) where a≈0.021, the curves are tangent. Also, solving the system may give other intersection points, but tangency is still satisfied at that point.Therefore, the answer should include all four α's: 2/3, 3/2, (13 -√601)/12, and (13 +√601)/12.But let me check if maybe these two irrational α's can be written in a simplified form or if they are actually equal to 2/3 and 3/2. Wait, compute (13 -√601)/12 numerically:√601≈24.51, so (13 -24.51)/12≈-11.51/12≈-0.959.Similarly, (13 +√601)/12≈(13 +24.51)/12≈37.51/12≈3.125. These are different from 2/3≈0.666 and 3/2=1.5.Therefore, the four solutions are distinct. Therefore, the answer is four values of α.But wait, the problem might have a mistake here. Let me check with another method. Maybe find the discriminant of the system.Alternatively, to find the values of α for which the system of equations has exactly one solution (double root). But since tangency can occur at multiple points, perhaps the system has either one or more solutions, but with at least one having multiplicity.Alternatively, substitute one equation into the other and set discriminant to zero for tangency.Let me try substituting y from the first equation into the second.Given:y = αx² + αx +1/24 ... (1)x = αy² + αy +1/24 ... (2)Substitute y from (1) into (2):x = α[αx² + αx +1/24]^2 + α[αx² + αx +1/24] +1/24This is a quartic equation in x. For the curves to be tangent, this equation must have a double root. That is, the quartic must have a repeated root. The condition for a quartic to have a double root is that its discriminant is zero, but calculating the discriminant of a quartic is extremely complicated.Alternatively, since we know that if the curves are tangent at (a, a), then x=a is a double root. Therefore, the quartic equation would have (x -a)^2 as a factor. But since we have symmetry, maybe x=a is a root of multiplicity two.Alternatively, perhaps consider that when we substitute y into x, the quartic equation can be written as f(x) =0, and for tangency at x=a, f(a)=0 and f’(a)=0.Therefore, maybe using this approach.Let me denote f(x) = α[αx² + αx +1/24]^2 + α[αx² + αx +1/24] +1/24 -x =0Then, at the tangent point x=a, f(a)=0 and f’(a)=0.But this seems complicated, but maybe manageable.But let's note that when a =b, which we established is necessary, then x=a is a solution. So let's write the quartic equation and see if x=a is a double root.But perhaps better to use resultants or another method. Alternatively, since we've already found four solutions via the a =b approach, and they all check out numerically, perhaps they are indeed the correct answers.Alternatively, recall that when we derived under a =b, we found four solutions. However, when we check the problem's original equations, maybe there are only two solutions. Let me test with α=2/3 and 3/2.For α=2/3, the curves are tangent at (1/4,1/4). If we plot them, they touch at that point and have the same slope. Similarly, for α=3/2, tangent at (-1/6, -1/6). For the other α's, the curves also touch at symmetric points. Therefore, all four α's are valid.But maybe the problem expects only two solutions. Wait, but maybe in the process, we considered both α positive and negative, but (13 +√601)/12≈3.125 and (13 -√601)/12≈-0.96 are separate solutions. Hence, the answer should include all four.But to check, let's see if there's any other method.Alternatively, suppose we consider the curves as parabolas. The first is a parabola opening upward if α>0, downward if α<0. The second is a parabola opening to the right if α>0, to the left if α<0. For them to be tangent, they must touch at a single point with the slope matching. Depending on the orientation, there can be different configurations.But given the previous analysis and the four solutions found, I think the correct answer is all four values: 2/3, 3/2, (13 -√601)/12, and (13 +√601)/12.But let me check for α= (13 -√601)/12≈-0.96:The first parabola opens downward (since α<0), and the second parabola opens to the left (since α<0). A downward-opening parabola and a left-opening parabola can intersect tangentially. Similarly for α=(13 +√601)/12≈3.125, both parabolas open upward and to the right, and can intersect tangentially.Hence, these are valid.Therefore, the answer is four values of α: 2/3, 3/2, (13 -√601)/12, and (13 +√601)/12.But the problem says "Find all values of α", so unless there is a restriction to positive α or something, but the problem doesn't specify. Therefore, all four are valid.But wait, the problem might expect the answer in a simplified radical form. So:α=2/3, 3/2, (13 -√601)/12, and (13 +√601)/12.Alternatively, write the irrational ones as (-√601 +13)/12 and (√601 +13)/12.Therefore, box all four solutions.But let me verify with another approach. Let me consider substituting equation (1) into equation (2):x = αy² + αy +1/24But y = αx² + αx +1/24, so substitute:x = α(αx² + αx +1/24)^2 + α(αx² + αx +1/24) +1/24Let me expand this:Let me denote u=αx² +αx +1/24, then equation becomes x=αu² +αu +1/24But u=αx² +αx +1/24, so:x=α(αx² +αx +1/24)^2 +α(αx² +αx +1/24) +1/24Let me expand term by term:First term: α*(αx² +αx +1/24)^2Let me compute (αx² +αx +1/24)^2:= α²x^4 + 2α²x^3 + [2α*(1/24) + α²]x² + 2α*(1/24)x + (1/24)^2But expanding:(αx² +αx +1/24)^2 = α²x^4 + 2α²x^3 + (α² + (2α)(1/24))x² + (2α*(1/24))x + (1/24)^2= α²x^4 + 2α²x³ + (α² + α/12)x² + (α/12)x + 1/576Multiply by α:First term: α*(above) = α³x^4 + 2α³x³ + α(α² + α/12)x² + α*(α/12)x + α/576= α³x^4 + 2α³x³ + (α³ + α²/12)x² + α²/12 x + α/576Second term: α*(αx² +αx +1/24) = α²x² + α²x + α/24Third term: 1/24Therefore, entire equation:x = [α³x^4 + 2α³x³ + (α³ + α²/12)x² + α²/12 x + α/576] + [α²x² + α²x + α/24] +1/24Combine like terms:Left side: xRight side:α³x^4 +2α³x³ + [α³ + α²/12 + α²]x² + [α²/12 + α²]x + [α/576 + α/24 +1/24]Simplify each term:For x^4: α³For x^3: 2α³For x²: α³ + α²/12 + α² = α³ + (1/12 +1)α² = α³ + (13/12)α²For x terms: α²/12 + α² = (1/12 +1)α² = (13/12)α²Constants: α/576 + α/24 +1/24 = α(1/576 +1/24) +1/24 = α(1/576 +24/576) +1/24 = α(25/576) +1/24Therefore, the equation becomes:α³x^4 +2α³x³ + (α³ +13α²/12)x² + (13α²/12)x + (25α/576 +1/24) -x =0Bring x to the right:α³x^4 +2α³x³ + (α³ +13α²/12)x² + (13α²/12 -1)x + (25α/576 +1/24)=0This is a quartic equation. For the curves to be tangent, this equation must have a double root. The condition for a quartic to have a double root is that its discriminant is zero, but as mentioned before, this is very complicated.Alternatively, if we suppose that x=a is a double root, then both the quartic and its derivative are zero at x=a. This gives two equations to solve for a and α.But this approach is quite involved. However, since we already found solutions using the a =b method, and verified them numerically, maybe those are the only solutions. But given that the quartic could have other double roots where a ≠b, but our previous analysis suggested that such solutions can't exist, leading to a contradiction unless a =b. Therefore, the only double roots of the quartic must satisfy a =b. Therefore, the four α's found earlier are the only solutions.Therefore, the answer is α=2/3, 3/2, (13 -√601)/12, and (13 +√601)/12.But let's check if (13 -√601)/12 and (13 +√601)/12 can be simplified or expressed in terms of other radicals. Since 601 is a prime number (yes, 601 ÷ 2=300.5, ÷3≈200.333, etc., up to sqrt(601)≈24.5, and none divide 601), so √601 can't be simplified. Hence, these are the simplest forms.Therefore, the values of α are:α=2/3, α=3/2, α=(13 -√601)/12, and α=(13 +√601)/12.Hence, boxed answers:boxed{dfrac{2}{3}}, boxed{dfrac{3}{2}}, boxed{dfrac{13 - sqrt{601}}{12}}, boxed{dfrac{13 + sqrt{601}}{12}}But wait, the problem says "Find all values of α". Maybe it expects the answer in a particular form. However, given the problem's wording and the process, these four solutions are correct.But I recall that when α=2/3 and 3/2, the curves are tangent at (1/4,1/4) and (-1/6,-1/6). The other two α's correspond to tangent points with irrational coordinates. Therefore, all four are valid.But let me check with another perspective. Suppose we set the two curves equal and solve for intersections. If they are tangent, the system has exactly one solution (but in reality, quartic can have two real solutions, one double and one simple, but tangency requires at least one double root). But the condition is not that the system has exactly one solution, but that the curves are tangent, which can occur even if there are other intersections. Therefore, the correct approach is not to look for when the system has a single solution, but when there is at least one point where they are tangent, which corresponds to the quartic having at least one double root. Thus, four solutions as found earlier.Therefore, the final answer is all four values of α.Final AnswerThe values of ( alpha ) are (boxed{dfrac{2}{3}}), (boxed{dfrac{3}{2}}), (boxed{dfrac{13 - sqrt{601}}{12}}), and (boxed{dfrac{13 + sqrt{601}}{12}}).</think>To find all values of ( alpha ) for which the curves ( y = alpha x^2 + alpha x + frac{1}{24} ) and ( x = alpha y^2 + alpha y + frac{1}{24} ) are tangent to each other, we start by noting that tangency implies the curves intersect at a point ((a, b)) and have the same slope at that point. First, we substitute the point ((a, b)) into both equations:1. ( b = alpha a^2 + alpha a + frac{1}{24} )2. ( a = alpha b^2 + alpha b + frac{1}{24} )Next, we compute the derivatives of both curves at the point ((a, b)):- For the first curve, the derivative is ( frac{dy}{dx} = 2alpha a + alpha ).- For the second curve, the derivative is ( frac{dy}{dx} = frac{1}{alpha (2b + 1)} ).Setting these derivatives equal gives:[ 2alpha a + alpha = frac{1}{alpha (2b + 1)} ]Multiplying both sides by ( alpha (2b + 1) ) results in:[ (2a + 1) alpha^2 (2b + 1) = 1 ]Assuming ( a = b ) (due to the symmetry of the equations), we substitute ( a ) for ( b ) in the equations:1. ( a = alpha a^2 + alpha a + frac{1}{24} )2. ( (2a + 1)^2 alpha^2 = 1 )Solving these equations, we find:- For ( alpha = frac{1}{2a + 1} ), we get the quadratic equation ( 24a^2 - 2a - 1 = 0 ), leading to ( a = frac{1}{4} ) or ( a = -frac{1}{6} ). This gives ( alpha = frac{2}{3} ) or ( alpha = frac{3}{2} ).- For ( alpha = -frac{1}{2a + 1} ), we get the quadratic equation ( 72a^2 + 46a - 1 = 0 ), leading to ( a = frac{-23 pm sqrt{601}}{72} ). This gives ( alpha = frac{13 - sqrt{601}}{12} ) or ( alpha = frac{13 + sqrt{601}}{12} ).Thus, the values of ( alpha ) for which the curves are tangent are:[boxed{dfrac{2}{3}}, boxed{dfrac{3}{2}}, boxed{dfrac{13 - sqrt{601}}{12}}, boxed{dfrac{13 + sqrt{601}}{12}}],<think>Okay, so I need to find all values of α where the two curves y = αx² + αx + 1/24 and x = αy² + αy + 1/24 are tangent to each other. Hmm, tangency means they touch at a single point and share the same slope there. That should involve solving the equations simultaneously and setting their derivatives equal at that point. Let me start by writing down the equations:First curve: y = αx² + αx + 1/24Second curve: x = αy² + αy + 1/24Since tangency requires both the curves to meet at a point (x, y) and their derivatives dy/dx (slopes) to be equal there, I need to set up the equations accordingly.First, let's find the derivatives of both curves.For the first curve, differentiating y with respect to x:dy/dx = 2αx + αFor the second curve, since x is expressed in terms of y, maybe I should find dx/dy and then take the reciprocal to get dy/dx.Differentiating x with respect to y:dx/dy = 2αy + αTherefore, dy/dx for the second curve is 1/(2αy + α)So, at the point of tangency, the slopes must be equal:2αx + α = 1/(2αy + α)Also, the point (x, y) lies on both curves, so substituting y from the first equation into the second equation:x = αy² + αy + 1/24But y is equal to αx² + αx + 1/24. So substitute that in:x = α(αx² + αx + 1/24)² + α(αx² + αx + 1/24) + 1/24This seems complicated. Expanding that might lead to a higher-degree equation. Let me try to approach this step by step.Let me denote y = αx² + αx + 1/24. Then substituting into the second equation:x = αy² + αy + 1/24So substituting y gives:x = α(αx² + αx + 1/24)² + α(αx² + αx + 1/24) + 1/24Let me call this Equation (1). If I can solve this equation for x and α, then use the slope condition to get another equation, and solve them together.But this looks like a quartic equation in x, which might be difficult to handle. Maybe there is symmetry here because the equations look similar. Maybe x = y at the point of tangency? Let me check if that's possible.Suppose x = y. Then substituting into the first equation:x = αx² + αx + 1/24Rearranged: αx² + αx + 1/24 - x = 0Simplify: αx² + (α - 1)x + 1/24 = 0Similarly, substituting into the second equation (since x = y):x = αx² + αx + 1/24Which is the same equation. So if x = y, then both equations reduce to the same quadratic equation. Therefore, if x = y, then the curves intersect at points satisfying this quadratic. For tangency, we would need this quadratic equation to have exactly one solution, which means discriminant is zero.So discriminant D of αx² + (α - 1)x + 1/24 = 0 is:D = [ (α - 1) ]² - 4 * α * (1/24) = (α² - 2α + 1) - (4α / 24) = α² - 2α + 1 - (α / 6)Set D = 0:α² - 2α + 1 - (α / 6) = 0Multiply both sides by 6 to eliminate fraction:6α² - 12α + 6 - α = 0Simplify:6α² - 13α + 6 = 0Solving this quadratic:α = [13 ± sqrt(169 - 144)] / 12 = [13 ± sqrt(25)] / 12 = [13 ± 5]/12Thus, α = (13 + 5)/12 = 18/12 = 3/2 or α = (13 - 5)/12 = 8/12 = 2/3So possible α values are 3/2 and 2/3. But I need to check if these actually correspond to points where the curves are tangent. Because we assumed x = y, but maybe there are other points where x ≠ y which could also be tangent. So I need to verify if these are the only solutions or if there are more.Alternatively, maybe the only possible tangent points are where x = y, given the symmetry. Let's check these solutions first.First, take α = 3/2. Then the quadratic equation becomes (3/2)x² + (3/2 - 1)x + 1/24 = 0 → (3/2)x² + (1/2)x + 1/24 = 0.Multiply by 24 to eliminate denominators:24*(3/2)x² + 24*(1/2)x + 24*(1/24) = 0 → 36x² + 12x + 1 = 0Discriminant D = 144 - 144 = 0. So this has a repeated root. The root is x = [-12]/(2*36) = -12/72 = -1/6. So x = y = -1/6. So the point is (-1/6, -1/6). Let's check if the slopes are equal here.First curve: dy/dx = 2αx + α. For α=3/2, x=-1/6:2*(3/2)*(-1/6) + 3/2 = 3*(-1/6) + 3/2 = -1/2 + 3/2 = 1Second curve: dy/dx = 1/(2αy + α). For α=3/2, y=-1/6:1/(2*(3/2)*(-1/6) + 3/2) = 1/( (3*(-1/6)) + 3/2 ) = 1/( -1/2 + 3/2 ) = 1/(1) = 1So slopes are equal here. So α=3/2 is a valid solution.Similarly, check α=2/3. Quadratic equation becomes (2/3)x² + (2/3 -1)x +1/24 =0 → (2/3)x² - (1/3)x +1/24=0Multiply by 24: 16x² - 8x +1=0Discriminant D=64 - 64=0. So repeated root. x=8/(2*16)=8/32=1/4. So x=y=1/4. Check slopes.First curve: dy/dx=2*(2/3)*(1/4)+2/3= (4/3)*(1/4) +2/3=1/3 +2/3=1Second curve: dy/dx=1/(2*(2/3)*(1/4)+2/3)=1/( (4/3)*(1/4) +2/3 )=1/(1/3 +2/3)=1/1=1Slopes are equal here as well. So α=2/3 is also valid.But now, could there be other points where x ≠ y and the curves are tangent? Let's check.Suppose that at the tangent point, x ≠ y. Then we have to solve the original equations without assuming x=y. Let's attempt that.We have two equations:1) y = αx² + αx + 1/242) x = αy² + αy + 1/24And the slope condition:2αx + α = 1/(2αy + α)Let me denote m = 2αx + α. Then the slope condition is m = 1/( (2αy + α) ). Let's note that m = 2αx + α and 2αy + α = 1/m.Therefore, 2αy + α = 1/m ⇒ 2αy = 1/m - α ⇒ y = (1/(2α m) - 1/2 )But m = 2αx + α, so substitute m into the expression for y:y = [1/(2α(2αx + α)) ] - 1/2 = [1/(2α*(α(2x +1)) ) ] -1/2 = [1/(2α²(2x +1)) ] -1/2But from equation 1, y is also equal to αx² + αx +1/24. So set them equal:αx² + αx +1/24 = [1/(2α²(2x +1)) ] -1/2This seems complicated, but maybe we can multiply both sides by 2α²(2x +1):2α²(2x +1)(αx² + αx +1/24) = 1 - α²(2x +1)This will lead to a polynomial equation in x. Let's compute left side:2α²(2x +1)(αx² + αx +1/24)First, expand (2x +1)(αx² + αx +1/24):Multiply term by term:2x*(αx²) + 2x*(αx) + 2x*(1/24) +1*(αx²) +1*(αx) +1*(1/24)= 2αx³ + 2αx² + (2x)/24 + αx² + αx + 1/24Simplify:2αx³ + (2αx² + αx²) + ( (2x)/24 + αx ) + 1/24= 2αx³ + 3αx² + ( (x)/12 + αx ) + 1/24Combine like terms:x terms: (1/12 + α)xSo overall:2αx³ + 3αx² + ( (1/12 + α )x ) + 1/24Multiply by 2α²:Left side becomes:2α² * [2αx³ + 3αx² + ( (1/12 + α )x ) + 1/24 ]= 2α²*(2αx³) + 2α²*(3αx²) + 2α²*( (1/12 + α )x ) + 2α²*(1/24 )= 4α³x³ + 6α³x² + 2α²(1/12 + α )x + (2α²)/24Simplify each term:First term: 4α³x³Second term: 6α³x²Third term: 2α²*(1/12 + α )x = (2α²/12 + 2α³)x = (α²/6 + 2α³)xFourth term: (2α²)/24 = α²/12Therefore, left side is:4α³x³ + 6α³x² + (α²/6 + 2α³)x + α²/12Right side is 1 - α²(2x +1) = 1 - 2α²x - α²So set left side equal to right side:4α³x³ + 6α³x² + (α²/6 + 2α³)x + α²/12 = 1 - 2α²x - α²Bring all terms to left side:4α³x³ + 6α³x² + (α²/6 + 2α³)x + α²/12 -1 + 2α²x + α² =0Simplify term by term:The x³ term: 4α³x³The x² term:6α³x²The x term: (α²/6 + 2α³ + 2α²)xConstants: α²/12 -1 + α² = (α²/12 + α²) -1 = (13α²/12 -1)So the equation becomes:4α³x³ + 6α³x² + [2α³ + (α²/6 + 2α²)]x + (13α²/12 -1 ) =0Wait, let me check the x term again:Original left side x term: (α²/6 + 2α³)xRight side brought over: +2α²xSo total x term: (α²/6 + 2α³ + 2α²)xFactor terms:= (2α³ + 2α² + α²/6)x = (2α³ + (12α²/6 + α²/6))x = (2α³ +13α²/6)xSimilarly, constants:α²/12 + α² -1 = (1α² +12α²)/12 -1 = 13α²/12 -1Thus, the equation is:4α³x³ +6α³x² + (2α³ +13α²/6)x +13α²/12 -1 =0This is a cubic equation in x. Solving this for general α is difficult, but maybe we can factor this equation if we substitute the known solutions α=2/3 and α=3/2.First, let's check for α=2/3.Substitute α=2/3:First term: 4*(8/27)x³ = 32/27 x³Second term:6*(8/27)x² = 48/27 x²Third term: [2*(8/27) +13*(4/9)/6 ]x = [16/27 + (52/9)/6 ]x = [16/27 + 52/54]x = [16/27 +26/27]x = 42/27 xFourth term:13*(4/9)/12 -1 = (52/9)/12 -1 = 52/108 -1 = 13/27 -1 = -14/27So equation becomes:32/27 x³ +48/27 x² +42/27 x -14/27 =0Multiply through by 27:32x³ +48x² +42x -14=0We know that when α=2/3, x=1/4 is a solution (from earlier). Let's check if x=1/4 is a root:32*(1/4)^3 +48*(1/4)^2 +42*(1/4) -14=32*(1/64) +48*(1/16) +42*(1/4) -14=0.5 +3 +10.5 -14 = 0.5+3=3.5; 3.5+10.5=14; 14-14=0. Yes, x=1/4 is a root.Therefore, (x -1/4) is a factor. Let's perform polynomial division.Divide 32x³ +48x² +42x -14 by (x -1/4). Alternatively, use synthetic division with x=1/4.Multiply coefficients by 1/4 each step:Coefficients: 32 | 48 | 42 | -14Bring down 32Multiply by 1/4: 8Add to next coefficient:48+8=56Multiply by1/4:14Add to next coefficient:42+14=56Multiply by1/4:14Add to last coefficient:-14+14=0Thus, the quotient is 32x² +56x +56. So the cubic factors as (x -1/4)(32x² +56x +56)=0. The quadratic 32x² +56x +56=0 has discriminant 3136 - 7168= -4032 <0, so no real roots. Therefore, for α=2/3, the only real solution is x=1/4, which we already considered. So that's the tangent point.Similarly, check for α=3/2.Substitute α=3/2 into the cubic equation:First term:4*(27/8)x³= (108/8)x³=13.5x³Second term:6*(27/8)x²=162/8x²=20.25x²Third term:[2*(27/8) +13*(9/4)/6 ]x = [54/8 + (117/4)/6 ]x = [54/8 +117/24 ]x = [162/24 +117/24]x=279/24x=11.625xFourth term:13*(9/4)/12 -1= (117/4)/12 -1=117/48 -1= (117 -48)/48=69/48=23/16≈1.4375Wait, but this contradicts the previous calculation for α=3/2. Wait, let's compute the constants correctly.Wait, α=3/2:Fourth term:13α²/12 -1=13*(9/4)/12 -1= (117/48) -1= (117 -48)/48=69/48=23/16≈1.4375So equation becomes:13.5x³ +20.25x² +11.625x +23/16=0But when α=3/2, we found x=-1/6 is a solution. Let's check x=-1/6:13.5*(-1/6)^3 +20.25*(-1/6)^2 +11.625*(-1/6) +23/16First term:13.5*(-1/216)= -13.5/216= -0.0625Second term:20.25*(1/36)=20.25/36≈0.5625Third term:11.625*(-1/6)≈-1.9375Fourth term:23/16≈1.4375Adding up: -0.0625 +0.5625=0.5; 0.5 -1.9375= -1.4375; -1.4375 +1.4375=0. So x=-1/6 is a root.Therefore, factor out (x +1/6):Using synthetic division with x=-1/6:Coefficients:13.5 |20.25 |11.625 |23/16≈1.4375Convert to fractions for precision:13.5=27/2, 20.25=81/4, 11.625=93/8, 23/16Bring down 27/2Multiply by -1/6: -27/12= -9/4Add to next coefficient:81/4 + (-9/4)=72/4=18Multiply by -1/6: -18/6= -3Add to next coefficient:93/8 + (-3)=93/8 -24/8=69/8Multiply by -1/6: -69/48= -23/16Add to last term:23/16 + (-23/16)=0Thus, the quotient is 27/2 x² +18x +69/8. Which is equivalent to:27/2 x² +18x +69/8= (108x² +144x +69)/8. The quadratic equation 108x² +144x +69=0. Compute discriminant:D=144² -4*108*69=20736 -4*108*69Calculate 4*108=432, 432*69=432*70 -432=30240 -432=29808So D=20736 -29808= -9072 <0. So no real roots. Hence, only x=-1/6 is the real root. So again, only the point x=y is the solution.Therefore, seems like the only tangent points are when x=y, which gives α=2/3 and 3/2. But just to be thorough, we need to check if there are other solutions where x ≠ y. How?Suppose there exists a solution where x ≠ y. Then the cubic equation we derived must have a real root other than x=1/4 or x=-1/6 for α=2/3 and 3/2, but maybe for other α?Wait, but for the cubic equation in x, which is:4α³x³ +6α³x² + (2α³ +13α²/6)x +13α²/12 -1 =0For this cubic to have a real root (x), perhaps for some α≠2/3,3/2, there could be another tangent point where x ≠ y. To check this, maybe consider that the curves could intersect at two points but tangent at one, but since the problem says "are tangent to each other," which might mean they are tangent (i.e., meet at exactly one point with matching slopes). So maybe in addition to the x=y solutions, there could be cases where two curves intersect at two points but tangent at one of them, but the problem requires that all points of intersection are tangent? Wait, the problem states "the curves ... are tangent to each other." So tangency could occur at one or more points, but if they intersect at more than one point, even if tangent at one, they wouldn't be considered as "tangent to each other" overall. So probably, the curves must have exactly one intersection point where they are tangent. Therefore, the system of equations must have exactly one solution, which occurs when the quartic equation (from substituting y into x) has a multiple root. However, solving quartic equations is complicated.Alternatively, maybe there are only two possible α where curves are tangent, which are the ones found with x=y.Alternatively, maybe another approach is better. Let me think.Let me consider the system:y = αx² + αx +1/24x = αy² + αy +1/24If I swap x and y, the equations transform into each other. Therefore, the system is symmetric under swapping x and y. Therefore, the solutions are either on the line x=y or come in pairs (x,y) and (y,x). But for tangency, if the curves are tangent at a point not on x=y, then there must be another tangent point at (y,x), but tangency requires only one point, so such points would have to coincide, meaning x=y. Hence, probably the only tangent points are on x=y. Therefore, the only possible α are the ones found earlier, 2/3 and 3/2.Therefore, the answer is α=2/3 and α=3/2.But let me check with an example. Take α=1. The curves are y=x²+x+1/24 and x=y²+y+1/24. Are these tangent? Let's see.Solve y = x² +x +1/24 and x = y² +y +1/24. Let's substitute y from first equation into second:x = (x² +x +1/24)^2 + (x² +x +1/24) +1/24Expand:x = (x^4 + 2x³ + (1/12 +1)x² + (1/12)x +1/576 ) +x² +x +1/24 +1/24Wait, this seems messy. Let me compute (x² +x +1/24)^2:= x^4 + 2x³ + (1 + 2/24)x² + (2x)(1/24) + (1/24)^2Wait, actually:(a + b + c)^2 = a² + b² + c² +2ab +2ac +2bcSo (x² +x +1/24)^2 =x^4 +x² + (1/24)^2 + 2x³ +2x²*(1/24) +2x*(1/24)= x^4 + 2x³ + (1 + 1/12)x² + (1/12)x +1/576Then, adding (x² +x +1/24):x^4 + 2x³ + (1 +1/12)x² + (1/12)x +1/576 +x² +x +1/24Combine like terms:x^4 +2x³ + [1 +1/12 +1]x² + [1/12 +1]x + [1/576 +1/24]= x^4 +2x³ + (2 +1/12)x² + (13/12)x + (1/576 +24/576)= x^4 +2x³ +25/12x² +13/12x +25/576So equation is x = x^4 +2x³ +25/12x² +13/12x +25/576Bring x to left:x^4 +2x³ +25/12x² +13/12x -x +25/576=0Simplify x terms:13/12x -x =13/12x -12/12x=1/12xThus equation:x^4 +2x³ +25/12x² +1/12x +25/576=0Multiply through by 576 to eliminate denominators:576x^4 +1152x³ +1200x² +48x +25=0This quartic equation. Let me check if it has real roots. Using rational root theorem, possible roots are ±1, ±5, etc. over factors of 576. Seems complicated. Maybe it has no real roots. For α=1, maybe the curves do not intersect, so not tangent. Hence, confirming that α=1 is not a solution. Hence, only α=2/3 and 3/2 are solutions.Therefore, the answer is α=2/3 and α=3/2.Final AnswerThe values of ( alpha ) are ( boxed{dfrac{2}{3}} ) and ( boxed{dfrac{3}{2}} ).</think>To find all values of ( alpha ) for which the curves ( y = alpha x^2 + alpha x + frac{1}{24} ) and ( x = alpha y^2 + alpha y + frac{1}{24} ) are tangent to each other, we start by noting that tangency requires the curves to meet at a single point with equal slopes at that point.First, we assume ( x = y ) at the point of tangency. Substituting ( y = x ) into both equations, we get:[ x = alpha x^2 + alpha x + frac{1}{24} ]Rearranging this equation, we have:[ alpha x^2 + (alpha - 1)x + frac{1}{24} = 0 ]For this quadratic equation to have exactly one solution (a repeated root), its discriminant must be zero. The discriminant ( D ) is given by:[ D = (alpha - 1)^2 - 4 alpha cdot frac{1}{24} ]Setting ( D = 0 ):[ (alpha - 1)^2 - frac{alpha}{6} = 0 ]Expanding and simplifying:[ alpha^2 - 2alpha + 1 - frac{alpha}{6} = 0 ]Multiplying through by 6 to clear the fraction:[ 6alpha^2 - 13alpha + 6 = 0 ]Solving this quadratic equation using the quadratic formula:[ alpha = frac{13 pm sqrt{169 - 144}}{12} = frac{13 pm 5}{12} ]This gives the solutions:[ alpha = frac{18}{12} = frac{3}{2} quad text{and} quad alpha = frac{8}{12} = frac{2}{3} ]We verify these solutions by checking the slopes at the points of tangency. For both ( alpha = frac{3}{2} ) and ( alpha = frac{2}{3} ), the slopes of the curves at the tangent points are equal, confirming tangency.Considering the symmetry of the equations and the requirement for tangency, we conclude that the only values of ( alpha ) for which the curves are tangent are ( frac{2}{3} ) and ( frac{3}{2} ).Thus, the values of ( alpha ) are ( boxed{dfrac{2}{3}} ) and ( boxed{dfrac{3}{2}} ).]

🔑:To find all values of alpha for which the curves y = alpha x^2 + alpha x + frac{1}{24} and x = alpha y^2 + alpha y + frac{1}{24} are tangent to each other, we need to consider the point of tangency (p, q) and ensure that the curves have the same slope at this point.1. Set up the system of equations: [ q = alpha p^2 + alpha p + frac{1}{24} quad cdots [1] ] [ p = alpha q^2 + alpha q + frac{1}{24} quad cdots [2] ]2. Subtract the equations: [ q - p = (alpha p^2 + alpha p + frac{1}{24}) - (alpha q^2 + alpha q + frac{1}{24}) ] [ q - p = alpha (p^2 - q^2) + alpha (p - q) ] [ q - p = alpha (p - q)(p + q) + alpha (p - q) ] [ (p - q){(p + q)alpha + alpha + 1} = 0 quad cdots [3] ]3. Case 1: q neq p [ (p + q)alpha + alpha + 1 = 0 ] [ p + q = -frac{alpha + 1}{alpha} quad cdots [4] ]4. Add the equations: [ q + p = alpha (p^2 + q^2) + alpha (p + q) + frac{1}{12} ] [ 2q = alpha (p^2 + q^2) + alpha (p + q) + frac{1}{12} ] [ 2alpha pq = (p + q){alpha (p + q) + alpha - 1} + frac{1}{12} ]5. Substitute [4] into the equation: [ pq = frac{25alpha + 24}{24alpha^2} quad cdots [5] ]6. Form the quadratic equation: [ t^2 + frac{alpha + 1}{alpha}t + frac{25alpha + 24}{24alpha^2} = 0 quad cdots [*] ]7. Condition for double root: [ left(frac{alpha + 1}{alpha}right)^2 - 4 cdot frac{25alpha + 24}{24alpha^2} = 0 ] [ left(frac{alpha + 1}{alpha}right)^2 = frac{4(25alpha + 24)}{24alpha^2} ] [ frac{(alpha + 1)^2}{alpha^2} = frac{100alpha + 96}{24alpha^2} ] [ (alpha + 1)^2 = frac{100alpha + 96}{24} ] [ 24(alpha^2 + 2alpha + 1) = 100alpha + 96 ] [ 24alpha^2 + 48alpha + 24 = 100alpha + 96 ] [ 24alpha^2 - 52alpha - 72 = 0 ] [ 6alpha^2 - 13alpha - 18 = 0 ] Solving this quadratic equation: [ alpha = frac{13 pm sqrt{601}}{12} ]8. Case 2: q = p [ p = alpha p^2 + alpha p + frac{1}{24} ] [ alpha p^2 + (alpha - 1)p + frac{1}{24} = 0 ] The discriminant must be zero for a double root: [ (alpha - 1)^2 - 4alpha cdot frac{1}{24} = 0 ] [ (alpha - 1)^2 = frac{4alpha}{24} ] [ (alpha - 1)^2 = frac{alpha}{6} ] [ 6(alpha - 1)^2 = alpha ] [ 6alpha^2 - 13alpha + 6 = 0 ] Solving this quadratic equation: [ alpha = frac{2}{3}, frac{3}{2} ]The final answer is boxed{frac{13 pm sqrt{601}}{12}, frac{2}{3}, frac{3}{2}}

🧠:[<think>Okay, so I have this problem here where I need to find the value of the expression ( a^{3} + sqrt{6} a^{2} + sqrt{2} a + sqrt{6} ) given that ( a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2} ). The answer choices are various roots, so probably I need to simplify the expression step by step. Let me try to work through this.First, let me write down the value of ( a ):( a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2} )Hmm, that looks a bit complicated. Maybe I can simplify the radical in the numerator? The term under the square root is ( 6 - 4 sqrt{2} ). I wonder if that can be expressed as a square of some expression. Let's see.Suppose ( 6 - 4 sqrt{2} = ( sqrt{a} - sqrt{b} )^2 ). Expanding the right side gives ( a + b - 2 sqrt{ab} ). Setting this equal to ( 6 - 4 sqrt{2} ), we have:1. ( a + b = 6 )2. ( 2 sqrt{ab} = 4 sqrt{2} Rightarrow sqrt{ab} = 2 sqrt{2} Rightarrow ab = 8 )So we need two numbers ( a ) and ( b ) such that their sum is 6 and product is 8. Let me solve these equations.From ( a + b = 6 ) and ( ab = 8 ), we can set up the quadratic equation ( x^2 - 6x + 8 = 0 ). Factoring this:( x^2 - 6x + 8 = (x - 2)(x - 4) = 0 )So the solutions are ( x = 2 ) and ( x = 4 ). Therefore, ( a = 4 ) and ( b = 2 ), or vice versa.Therefore, ( 6 - 4 sqrt{2} = ( sqrt{4} - sqrt{2} )^2 = (2 - sqrt{2})^2 ). Wait, let me check:( (2 - sqrt{2})^2 = 4 - 4 sqrt{2} + 2 = 6 - 4 sqrt{2} ). Yes, that works!So the square root simplifies to ( 2 - sqrt{2} ). Therefore, substituting back into the expression for ( a ):( a = frac{ -sqrt{6} + (2 - sqrt{2}) }{2} = frac{ 2 - sqrt{2} - sqrt{6} }{2 } )Hmm, so ( a = frac{2 - sqrt{2} - sqrt{6}}{2} ). Maybe we can split this fraction:( a = frac{2}{2} - frac{ sqrt{2} }{2 } - frac{ sqrt{6} }{2 } = 1 - frac{ sqrt{2} }{2 } - frac{ sqrt{6} }{2 } )But not sure if that helps yet. Alternatively, maybe factor out a negative sign? Let's see:But perhaps another approach. Let me think. The expression we need to compute is ( a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6} ). Maybe we can factor this expression or find a polynomial equation that ( a ) satisfies, so that we can substitute ( a^3 ), ( a^2 ), etc., with lower degree terms.So, if I can find a polynomial equation that ( a ) satisfies, then I can use that to reduce the powers of ( a ). Let's try to find such an equation.Given ( a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2} ). Let me first denote ( sqrt{6 - 4 sqrt{2}} = 2 - sqrt{2} ), as we found earlier. So:( a = frac{ -sqrt{6} + 2 - sqrt{2} }{2} )Let me rearrange terms:( a = frac{2 - sqrt{2} - sqrt{6}}{2} )Multiply both sides by 2:( 2a = 2 - sqrt{2} - sqrt{6} )Bring all terms to the left side:( 2a - 2 + sqrt{2} + sqrt{6} = 0 )Hmm, this might not be the most straightforward way. Let me consider squaring both sides to eliminate the square roots. Let me first isolate the square roots.From ( 2a = 2 - sqrt{2} - sqrt{6} ), let's rearrange:( sqrt{2} + sqrt{6} = 2 - 2a )Now, square both sides:Left side: ( (sqrt{2} + sqrt{6})^2 = 2 + 6 + 2 sqrt{12} = 8 + 4 sqrt{3} )Right side: ( (2 - 2a)^2 = 4 - 8a + 4a^2 )So:( 8 + 4 sqrt{3} = 4 - 8a + 4a^2 )Subtract 4 from both sides:( 4 + 4 sqrt{3} = -8a + 4a^2 )Divide both sides by 4:( 1 + sqrt{3} = -2a + a^2 )Rearrange:( a^2 - 2a - (1 + sqrt{3}) = 0 )Hmm, this is a quadratic equation in ( a ). So:( a^2 = 2a + 1 + sqrt{3} )This might be helpful. Let me note that.So, ( a^2 = 2a + 1 + sqrt{3} )But the original expression involves ( a^3 ). Let's compute ( a^3 ).Since ( a^3 = a cdot a^2 ). From above, ( a^2 = 2a + 1 + sqrt{3} ). Therefore:( a^3 = a cdot (2a + 1 + sqrt{3}) = 2a^2 + a + sqrt{3} a )But now we have ( a^3 ) in terms of ( a^2 ) and ( a ). But we can substitute ( a^2 ) again using the previous equation.( a^3 = 2(2a + 1 + sqrt{3}) + a + sqrt{3} a )Let me compute each term:First, ( 2(2a + 1 + sqrt{3}) = 4a + 2 + 2 sqrt{3} )Then, adding the remaining terms ( a + sqrt{3} a ):Total ( a^3 = 4a + 2 + 2 sqrt{3} + a + sqrt{3} a = (4a + a) + 2 + 2 sqrt{3} + sqrt{3} a )Simplify:( a^3 = 5a + 2 + 2 sqrt{3} + sqrt{3} a )Combine like terms:( a^3 = (5a + sqrt{3} a) + (2 + 2 sqrt{3}) )Factor out ( a ):( a^3 = a(5 + sqrt{3}) + 2(1 + sqrt{3}) )Hmm, not sure if this helps directly. Let's note the expression we need to compute:( a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6} )From above, ( a^3 = 5a + sqrt{3} a + 2 + 2 sqrt{3} ). Let's substitute that:Expression becomes:( [5a + sqrt{3} a + 2 + 2 sqrt{3}] + sqrt{6} a^2 + sqrt{2} a + sqrt{6} )Combine like terms:= ( sqrt{6} a^2 + (5a + sqrt{3} a + sqrt{2} a) + (2 + 2 sqrt{3} + sqrt{6}) )Factor out ( a ):= ( sqrt{6} a^2 + a(5 + sqrt{3} + sqrt{2}) + (2 + 2 sqrt{3} + sqrt{6}) )Hmm, still looks complicated. Maybe we can use the quadratic equation for ( a^2 ). From before, ( a^2 = 2a + 1 + sqrt{3} ). So, substitute ( a^2 ):= ( sqrt{6}(2a + 1 + sqrt{3}) + a(5 + sqrt{3} + sqrt{2}) + (2 + 2 sqrt{3} + sqrt{6}) )Compute each term:First term: ( sqrt{6}(2a + 1 + sqrt{3}) = 2 sqrt{6} a + sqrt{6} + sqrt{6} cdot sqrt{3} = 2 sqrt{6} a + sqrt{6} + sqrt{18} )Simplify ( sqrt{18} = 3 sqrt{2} ):So, first term becomes ( 2 sqrt{6} a + sqrt{6} + 3 sqrt{2} )Second term: ( a(5 + sqrt{3} + sqrt{2}) = 5a + sqrt{3} a + sqrt{2} a )Third term: ( 2 + 2 sqrt{3} + sqrt{6} )Now, combine all terms:First term: ( 2 sqrt{6} a + sqrt{6} + 3 sqrt{2} )Second term: ( 5a + sqrt{3} a + sqrt{2} a )Third term: ( 2 + 2 sqrt{3} + sqrt{6} )Combine all together:= (2 √6 a + 5a + √3 a + √2 a) + (√6 + 3 √2 + 2 + 2 √3 + √6)Let me group the terms with ( a ):= a (2 √6 + 5 + √3 + √2 ) + (√6 + 3 √2 + 2 + 2 √3 + √6 )Simplify the constants:For the ( a ) terms:Coefficient: ( 2 sqrt{6} + 5 + sqrt{3} + sqrt{2} ). Not sure if that can be simplified.For the constant terms:√6 + √6 = 2 √63 √2 remains2 remains2 √3 remainsSo total constants: 2 √6 + 3 √2 + 2 + 2 √3Hmm, this seems messy. Maybe there's a different approach. Let's go back to the original expression.Wait, perhaps instead of expanding all terms, maybe plugging in the value of ( a ) directly into the expression would work? Although that might involve a lot of computation, but since it's multiple choice, maybe some terms will cancel out.Alternatively, note that the given expression is ( a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6} ). Maybe factor by grouping?Let me see:Group ( a^3 + sqrt{6} a^2 ) and ( sqrt{2} a + sqrt{6} ):First group: ( a^2(a + sqrt{6}) )Second group: ( sqrt{2}(a + sqrt{3}) )... Wait, but ( sqrt{2} a + sqrt{6} = sqrt{2} a + sqrt{2} cdot sqrt{3} = sqrt{2}(a + sqrt{3}) )So the expression becomes:( a^2(a + sqrt{6}) + sqrt{2}(a + sqrt{3}) )Hmm, interesting. Now, if we can find a relationship between ( a + sqrt{6} ) and ( a + sqrt{3} ), maybe?Alternatively, let me check if ( a + sqrt{3} ) is related to previous expressions.Wait, earlier when we derived the quadratic equation for ( a ):We had ( a^2 - 2a - (1 + sqrt{3}) = 0 ), so ( a^2 = 2a + 1 + sqrt{3} )Therefore, ( a + sqrt{3} = a + sqrt{3} ). Not sure. Maybe substitute ( a^2 = 2a + 1 + sqrt{3} ) into the first term:First term: ( a^2(a + sqrt{6}) = (2a + 1 + sqrt{3})(a + sqrt{6}) )Let me expand this:= 2a(a) + 2a(sqrt{6}) + 1(a) + 1(sqrt{6}) + sqrt{3}(a) + sqrt{3}(sqrt{6})Simplify each term:= 2a^2 + 2a sqrt{6} + a + sqrt{6} + a sqrt{3} + sqrt{18}Again, ( sqrt{18} = 3 sqrt{2} ), so:= 2a^2 + 2a sqrt{6} + a + sqrt{6} + a sqrt{3} + 3 sqrt{2}Now, substitute ( a^2 = 2a + 1 + sqrt{3} ):= 2(2a + 1 + sqrt{3}) + 2a sqrt{6} + a + sqrt{6} + a sqrt{3} + 3 sqrt{2}Compute the first term:= 4a + 2 + 2 sqrt{3} + 2a sqrt{6} + a + sqrt{6} + a sqrt{3} + 3 sqrt{2}Combine like terms:Linear terms in a: 4a + a = 5aTerms with ( a sqrt{6} ): 2a sqrt{6}Terms with ( a sqrt{3} ): a sqrt{3}Constant terms: 2 + 2 sqrt{3} + sqrt{6} + 3 sqrt{2}So:First term after substitution becomes:5a + 2a sqrt{6} + a sqrt{3} + 2 + 2 sqrt{3} + sqrt{6} + 3 sqrt{2}Now, the entire expression is:First term (as above) + Second term (which is ( sqrt{2}(a + sqrt{3}) )):So total expression:5a + 2a sqrt{6} + a sqrt{3} + 2 + 2 sqrt{3} + sqrt{6} + 3 sqrt{2} + sqrt{2} a + sqrt{2} sqrt{3}Simplify term by term:First, let's collect all terms with ( a ):5a + 2a sqrt{6} + a sqrt{3} + sqrt{2} aFactor out ( a ):a(5 + 2 sqrt{6} + sqrt{3} + sqrt{2})Then, the constant terms:2 + 2 sqrt{3} + sqrt{6} + 3 sqrt{2} + sqrt{2} sqrt{3}Simplify ( sqrt{2} sqrt{3} = sqrt{6} ), so:2 + 2 sqrt{3} + sqrt{6} + 3 sqrt{2} + sqrt{6} = 2 + 2 sqrt{3} + 2 sqrt{6} + 3 sqrt{2}So putting it all together:= a(5 + 2 sqrt{6} + sqrt{3} + sqrt{2}) + (2 + 2 sqrt{3} + 2 sqrt{6} + 3 sqrt{2})Hmm, this still looks complicated, but maybe we can compute the coefficients numerically to see if something cancels out. Alternatively, perhaps there's a relationship we haven't used yet.Wait, let me recall the value of ( a ):( a = frac{2 - sqrt{2} - sqrt{6}}{2} ). Let's compute ( a ) numerically to approximate its value and then compute the expression numerically. Maybe that can help us guess the answer or check our work.Compute ( sqrt{2} approx 1.4142 ), ( sqrt{6} approx 2.4495 ).So numerator: 2 - 1.4142 - 2.4495 ≈ 2 - 1.4142 - 2.4495 ≈ 2 - 3.8637 ≈ -1.8637Then, ( a ≈ -1.8637 / 2 ≈ -0.93185 )So ( a ≈ -0.93185 )Now compute the expression ( a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6} )First, compute each term:1. ( a^3 ≈ (-0.93185)^3 ≈ -0.93185 * 0.93185 * 0.93185 ≈ approx -0.808 )2. ( sqrt{6} a^2 ≈ 2.4495 * (0.93185)^2 ≈ 2.4495 * 0.868 ≈ approx 2.4495 * 0.868 ≈ 2.127 )3. ( sqrt{2} a ≈ 1.4142 * (-0.93185) ≈ -1.319 )4. ( sqrt{6} ≈ 2.4495 )Add them all up:-0.808 + 2.127 - 1.319 + 2.4495 ≈First, -0.808 + 2.127 = 1.3191.319 - 1.319 = 00 + 2.4495 ≈ 2.4495Wait, ( sqrt{6} ≈ 2.4495 ), so the approximate value is ( sqrt{6} ). But looking at the answer choices: (D) ( sqrt{6} ). So maybe the answer is D?But wait, but when I approximated, I got approximately 2.4495, which is ( sqrt{6} ), but the answer choices include both ( sqrt{6} ) and ( sqrt{2} ), etc. However, given that in the problem statement the answer is one of the options, and my approximate calculation gives ( sqrt{6} ), but let me check if my calculation is correct.Wait, step by step:1. ( a ≈ -0.93185 )2. ( a^3 ≈ (-0.93185)^3 ≈ -0.93185 * 0.93185 = -0.868, then * -0.93185 ≈ 0.808 (but since it's negative cubed, remains negative: -0.808)Wait, actually: (-0.93185)^3 = (-0.93185)*(-0.93185) = 0.868, then * (-0.93185) ≈ -0.808. Correct.3. ( sqrt{6} ≈ 2.4495 ), ( a^2 ≈ (0.93185)^2 ≈ 0.868 ), so ( sqrt{6} a^2 ≈ 2.4495 * 0.868 ≈ 2.127 ). Correct.4. ( sqrt{2} a ≈ 1.4142 * (-0.93185) ≈ -1.319 ). Correct.5. ( sqrt{6} ≈ 2.4495 ). Correct.Sum: -0.808 + 2.127 -1.319 + 2.4495 ≈First, -0.808 + 2.127 = 1.3191.319 -1.319 = 00 + 2.4495 ≈ 2.4495 ≈ ( sqrt{6} ). So numerical approximation suggests the answer is D) ( sqrt{6} ). But let's check if this is accurate.But wait, maybe my approximation is missing something. Let's see:Wait, but in the original expression, when I approximated, the result was roughly ( sqrt{6} ), so option D. However, let's try to see if there is a way to do this algebraically.Wait, earlier steps led me to the expression:= a(5 + 2 sqrt{6} + sqrt{3} + sqrt{2}) + (2 + 2 sqrt{3} + 2 sqrt{6} + 3 sqrt{2})If we can substitute the value of ( a ) here:Given ( a = frac{2 - sqrt{2} - sqrt{6}}{2} ), let's substitute this into the expression:First, compute the coefficient of ( a ):5 + 2 sqrt{6} + sqrt{3} + sqrt{2}Multiply this by ( a ):= (5 + 2 sqrt{6} + sqrt{3} + sqrt{2}) * ( frac{2 - sqrt{2} - sqrt{6}}{2} )Then, add the constant terms:2 + 2 sqrt{3} + 2 sqrt{6} + 3 sqrt{2}This seems complicated, but let me compute the first part step by step.Let me denote:Let’s compute ( (5 + 2 sqrt{6} + sqrt{3} + sqrt{2})(2 - sqrt{2} - sqrt{6}) ), then divide by 2, and then add the constant terms.First, expand the product:Multiply each term in the first parenthesis by each term in the second:First, 5 * 2 = 105 * (-√2) = -5√25 * (-√6) = -5√6Then, 2√6 * 2 = 4√62√6 * (-√2) = -2√12 = -2 * 2√3 = -4√32√6 * (-√6) = -2 * 6 = -12Next, √3 * 2 = 2√3√3 * (-√2) = -√6√3 * (-√6) = -√18 = -3√2Then, √2 * 2 = 2√2√2 * (-√2) = -2√2 * (-√6) = -√12 = -2√3Now, let's write all these terms:10 -5√2 -5√6 +4√6 -4√3 -12 +2√3 -√6 -3√2 +2√2 -2 -2√3Now, let's combine like terms:Constants: 10 -12 -2 = -4√2 terms: -5√2 -3√2 +2√2 = (-5 -3 +2)√2 = (-6)√2√3 terms: -4√3 +2√3 -2√3 = (-4 +2 -2)√3 = (-4)√3√6 terms: -5√6 +4√6 -√6 = (-5 +4 -1)√6 = (-2)√6So the entire product is:-4 -6√2 -4√3 -2√6Then divide by 2:= (-4/2) - (6√2)/2 - (4√3)/2 - (2√6)/2= -2 -3√2 -2√3 -√6Now, add the constant terms from before:2 + 2√3 + 2√6 + 3√2So total expression:(-2 -3√2 -2√3 -√6) + (2 + 2√3 + 2√6 + 3√2)Combine like terms:Constants: -2 + 2 = 0√2 terms: -3√2 +3√2 = 0√3 terms: -2√3 +2√3 = 0√6 terms: -√6 +2√6 = √6Therefore, the entire expression simplifies to √6. So the answer is D) √6.Wait, that works out perfectly! So all the terms cancel out except for √6. Therefore, the correct answer is D) √6.Final Answerboxed{D}</think>Given ( a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2} ), we need to find the value of the expression ( a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6} ).First, we simplify the radical in the numerator. We find that ( 6 - 4 sqrt{2} ) can be expressed as ( (2 - sqrt{2})^2 ). Thus, we rewrite ( a ) as:[a = frac{ -sqrt{6} + (2 - sqrt{2}) }{2} = frac{2 - sqrt{2} - sqrt{6}}{2}]Next, we derive a polynomial equation that ( a ) satisfies. Starting from:[2a = 2 - sqrt{2} - sqrt{6}]we isolate the square roots and square both sides:[sqrt{2} + sqrt{6} = 2 - 2a]Squaring both sides, we get:[8 + 4 sqrt{3} = 4 - 8a + 4a^2]Simplifying, we find:[a^2 - 2a - (1 + sqrt{3}) = 0]Thus, ( a^2 = 2a + 1 + sqrt{3} ).Using this quadratic equation, we compute ( a^3 ):[a^3 = a cdot a^2 = a(2a + 1 + sqrt{3}) = 2a^2 + a + sqrt{3}a]Substituting ( a^2 = 2a + 1 + sqrt{3} ):[a^3 = 2(2a + 1 + sqrt{3}) + a + sqrt{3}a = 5a + sqrt{3}a + 2 + 2sqrt{3}]Substituting ( a^3 ), ( a^2 ), and ( a ) into the original expression:[a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}]we get:[(5a + sqrt{3}a + 2 + 2sqrt{3}) + sqrt{6}(2a + 1 + sqrt{3}) + sqrt{2}a + sqrt{6}]Expanding and simplifying each term, we combine like terms and find that all terms cancel out except for ( sqrt{6} ).Thus, the value of the expression is:[boxed{D}],<think>Okay, let's see. I need to find the value of the expression (a^{3} + sqrt{6} a^{2} + sqrt{2} a + sqrt{6}) where (a = frac{-sqrt{6} + sqrt{6 - 4 sqrt{2}}}{2}). The options are all in terms of square roots, so maybe simplifying (a) first would help?First, let me write down the given value of (a):[a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2}]Hmm, the expression inside the square root, (6 - 4 sqrt{2}), looks like it could be simplified. Maybe it's a square of some expression? Let me check.Suppose (6 - 4 sqrt{2}) can be written as ((sqrt{a} - sqrt{b})^2). Then expanding that would give (a + b - 2 sqrt{ab}). Comparing with (6 - 4 sqrt{2}), we have:[a + b = 6 quad text{and} quad -2 sqrt{ab} = -4 sqrt{2}]From the second equation: (2 sqrt{ab} = 4 sqrt{2}) => (sqrt{ab} = 2 sqrt{2}) => (ab = 8). So we have two equations:1. (a + b = 6)2. (ab = 8)Solving these equations. Let me denote (a) and (b) as the roots of the quadratic equation (x^2 - 6x + 8 = 0). Factoring:[x^2 - 6x + 8 = (x - 2)(x - 4) = 0]So, the solutions are (x = 2) and (x = 4). Therefore, (a = 4) and (b = 2), or vice versa. Therefore,[6 - 4 sqrt{2} = (sqrt{4} - sqrt{2})^2 = (2 - sqrt{2})^2]Wait, let's check that:[(2 - sqrt{2})^2 = 4 - 4 sqrt{2} + 2 = 6 - 4 sqrt{2}]Yes, that's correct! So, the square root simplifies to (2 - sqrt{2}). Therefore, substituting back into (a):[a = frac{ -sqrt{6} + (2 - sqrt{2}) }{2} = frac{2 - sqrt{2} - sqrt{6}}{2}]Simplify numerator:[a = frac{2 - sqrt{6} - sqrt{2}}{2} = 1 - frac{sqrt{6} + sqrt{2}}{2}]Hmm, not sure if that helps immediately. Maybe I should consider if (a) satisfies a certain quadratic equation? Since it's given in terms of square roots, perhaps squaring both sides could help find an equation that (a) satisfies. Let's try that.Start with the original expression for (a):[a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2}]Let me denote (b = sqrt{6 - 4 sqrt{2}} = 2 - sqrt{2}) as we found earlier. Therefore, (a = frac{ -sqrt{6} + (2 - sqrt{2}) }{2} )So, (a = frac{2 - sqrt{6} - sqrt{2}}{2}). Let me write this as:[2a = 2 - sqrt{6} - sqrt{2}]Bring all terms to the left side:[2a - 2 + sqrt{6} + sqrt{2} = 0]But this might not be the most straightforward approach. Let me instead square both sides of the original equation for (a) to eliminate the square root. Let's try that.Original equation:[2a + sqrt{6} = sqrt{6 - 4 sqrt{2}}]Wait, let me rearrange the original expression:[2a + sqrt{6} = sqrt{6 - 4 sqrt{2}}]Yes, because if (a = frac{ -sqrt{6} + sqrt{6 - 4 sqrt{2}} }{2}), then multiplying both sides by 2:[2a = -sqrt{6} + sqrt{6 - 4 sqrt{2}}]Then adding (sqrt{6}) to both sides:[2a + sqrt{6} = sqrt{6 - 4 sqrt{2}}]Now, square both sides:Left side: ((2a + sqrt{6})^2 = 4a^2 + 4a sqrt{6} + 6)Right side: ((sqrt{6 - 4 sqrt{2}})^2 = 6 - 4 sqrt{2})Set them equal:[4a^2 + 4a sqrt{6} + 6 = 6 - 4 sqrt{2}]Subtract 6 from both sides:[4a^2 + 4a sqrt{6} = -4 sqrt{2}]Divide both sides by 4:[a^2 + a sqrt{6} = -sqrt{2}]So, we get an equation:[a^2 + sqrt{6} a = -sqrt{2}]Wait, that's interesting. Let me note that down. The expression we need to compute is (a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}). Maybe we can use the equation above to simplify this expression.First, let's note that from the equation:[a^2 = -sqrt{6} a - sqrt{2}]So, we can substitute (a^2) in higher powers. Let's try to compute (a^3).Compute (a^3):[a^3 = a cdot a^2 = a (-sqrt{6} a - sqrt{2}) = -sqrt{6} a^2 - sqrt{2} a]But we can substitute (a^2) again using the equation:[a^3 = -sqrt{6} (-sqrt{6} a - sqrt{2}) - sqrt{2} a]Let me compute each term:First term: (-sqrt{6} (-sqrt{6} a) = (sqrt{6})^2 a = 6 a)Second term: (-sqrt{6} (-sqrt{2}) = sqrt{12} = 2 sqrt{3})Third term: (-sqrt{2} a)So, putting together:[a^3 = 6a + 2 sqrt{3} - sqrt{2} a = (6 - sqrt{2}) a + 2 sqrt{3}]Hmm, that introduces a (sqrt{3}) term, which is not present in the original expression. The original expression to compute is (a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}). Let me substitute (a^3) and (a^2):First, (a^3 = (6 - sqrt{2}) a + 2 sqrt{3})Second, (sqrt{6} a^2 = sqrt{6} (-sqrt{6} a - sqrt{2}) = -6 a - sqrt{12} = -6 a - 2 sqrt{3})Third, (sqrt{2} a) remains as is.Fourth, (sqrt{6}) is just added.Now, let's combine all terms:(a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}) becomes:[[(6 - sqrt{2}) a + 2 sqrt{3}] + [-6 a - 2 sqrt{3}] + sqrt{2} a + sqrt{6}]Let me compute term by term:1. Expand the first bracket: (6a - sqrt{2} a + 2 sqrt{3})2. Expand the second bracket: (-6a - 2 sqrt{3})3. Third term: (sqrt{2} a)4. Fourth term: (sqrt{6})Now, combine like terms:For (a) terms:(6a - sqrt{2} a -6a + sqrt{2} a = (6a -6a) + (-sqrt{2} a + sqrt{2} a) = 0)For (sqrt{3}) terms:(2 sqrt{3} - 2 sqrt{3} = 0)So, all that remains is (sqrt{6}). Therefore, the entire expression simplifies to (sqrt{6}). Therefore, the answer is D) (sqrt{6}).But let me check again to make sure I didn't make a mistake in the algebra.Starting with computing (a^3):We had (a^2 = -sqrt{6} a - sqrt{2}), so (a^3 = a cdot a^2 = a (-sqrt{6} a - sqrt{2}) = -sqrt{6} a^2 - sqrt{2} a). Then substituting (a^2) again:(-sqrt{6} (-sqrt{6} a - sqrt{2}) - sqrt{2} a = sqrt{6} cdot sqrt{6} a + sqrt{6} cdot sqrt{2} - sqrt{2} a = 6a + sqrt{12} - sqrt{2} a). Wait, (sqrt{12}) is (2 sqrt{3}), so that becomes (6a - sqrt{2} a + 2 sqrt{3}). So, (a^3 = (6 - sqrt{2})a + 2 sqrt{3}). That's correct.Then, (sqrt{6} a^2 = sqrt{6} (-sqrt{6} a - sqrt{2}) = - (sqrt{6} cdot sqrt{6}) a - sqrt{6} cdot sqrt{2} = -6a - sqrt{12} = -6a - 2 sqrt{3}). Correct.Adding up (a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}):[[(6 - sqrt{2})a + 2 sqrt{3}] + [-6a - 2 sqrt{3}] + sqrt{2} a + sqrt{6}]Combine (a) terms:- ((6 - sqrt{2})a -6a + sqrt{2} a = [6a -6a] + [ -sqrt{2} a + sqrt{2} a ] = 0 + 0 = 0)Combine (sqrt{3}) terms:(2 sqrt{3} -2 sqrt{3} = 0)Then the remaining term is (sqrt{6}). So yes, the entire expression simplifies to (sqrt{6}). Therefore, the answer is D.Alternatively, another approach could be to recognize that the expression to compute might be constructed from the minimal equation of (a). Since we found that (a^2 + sqrt{6} a = -sqrt{2}), we can perhaps use this to reduce the cubic term.Alternatively, maybe plugging in the value of (a) directly into the expression would work, but that might involve messy calculations. Let me try that briefly to confirm.Given (a = frac{2 - sqrt{6} - sqrt{2}}{2}). Let me compute (a^3), (a^2), etc. But that might be time-consuming. Alternatively, let me use decimal approximations to check.First, compute numerical values:Compute (sqrt{6} approx 2.449), (sqrt{2} approx 1.414), then:Compute (6 - 4 sqrt{2} approx 6 - 4*1.414 ≈ 6 - 5.656 ≈ 0.344). Then sqrt of that is approx 0.586. Then, (a ≈ (-2.449 + 0.586)/2 ≈ (-1.863)/2 ≈ -0.9315).So, a ≈ -0.9315. Now compute the expression:a³ + sqrt(6) a² + sqrt(2) a + sqrt(6)Compute each term:First, a ≈ -0.9315a³ ≈ (-0.9315)^3 ≈ -0.808sqrt(6) ≈ 2.449, a² ≈ (-0.9315)^2 ≈ 0.867. So sqrt(6)*a² ≈ 2.449*0.867 ≈ 2.123sqrt(2)*a ≈ 1.414*(-0.9315) ≈ -1.318sqrt(6) ≈ 2.449Adding all up: -0.808 + 2.123 -1.318 + 2.449 ≈ (-0.808 -1.318) + (2.123 + 2.449) ≈ (-2.126) + (4.572) ≈ 2.446. Since sqrt(6) ≈ 2.449, which is very close, considering the approximations, so likely exact value is sqrt(6). Therefore, answer D.But just to check, maybe my approximations led me to that. Alternatively, the exact calculation we did earlier shows it's exactly sqrt(6). So, confident the answer is D.Final Answerboxed{D}</think>Given (a = frac{-sqrt{6} + sqrt{6 - 4 sqrt{2}}}{2}), we need to find the value of the expression (a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}).First, we simplify the expression inside the square root:[6 - 4 sqrt{2} = (2 - sqrt{2})^2]Thus,[sqrt{6 - 4 sqrt{2}} = 2 - sqrt{2}]Substituting this back into (a), we get:[a = frac{-sqrt{6} + (2 - sqrt{2})}{2} = frac{2 - sqrt{6} - sqrt{2}}{2}]Next, we square both sides of the equation (2a + sqrt{6} = sqrt{6 - 4 sqrt{2}}) to find a quadratic equation satisfied by (a):[(2a + sqrt{6})^2 = (sqrt{6 - 4 sqrt{2}})^2]Simplifying both sides, we get:[4a^2 + 4asqrt{6} + 6 = 6 - 4sqrt{2}]Subtracting 6 from both sides and dividing by 4, we obtain:[a^2 + sqrt{6}a = -sqrt{2}]Using this quadratic equation, we substitute (a^2 = -sqrt{6}a - sqrt{2}) into the expression (a^3 + sqrt{6} a^2 + sqrt{2} a + sqrt{6}):[a^3 = a cdot a^2 = a(-sqrt{6}a - sqrt{2}) = -sqrt{6}a^2 - sqrt{2}a]Substituting (a^2 = -sqrt{6}a - sqrt{2}) again, we get:[a^3 = -sqrt{6}(-sqrt{6}a - sqrt{2}) - sqrt{2}a = 6a + sqrt{12} - sqrt{2}a = (6 - sqrt{2})a + 2sqrt{3}]Now, substituting (a^3) and (a^2) into the expression:[a^3 + sqrt{6}a^2 + sqrt{2}a + sqrt{6}]becomes:[[(6 - sqrt{2})a + 2sqrt{3}] + sqrt{6}(-sqrt{6}a - sqrt{2}) + sqrt{2}a + sqrt{6}]Simplifying term by term:[(6 - sqrt{2})a + 2sqrt{3} - 6a - 2sqrt{3} + sqrt{2}a + sqrt{6}]Combining like terms, we find all (a) terms and (sqrt{3}) terms cancel out, leaving:[sqrt{6}]Thus, the value of the expression is (boxed{D}).]

🔑:-、1.D.Notice that a=frac{-sqrt{6}+sqrt{sqrt{6}^{2}-4 times 1 times sqrt{2}}}{2 times 1}, then a is a root of the equation x^{2}+sqrt{6} x+sqrt{2}=0.Therefore, a^{2}+sqrt{6} a+sqrt{2}=0.Thus, a^{3}+sqrt{6} a^{2}+sqrt{2} a+sqrt{6}=aleft(a^{2}+sqrt{6} a+sqrt{2}right)+sqrt{6}=sqrt{6} .

🧠:[<think>Okay, let's see. The problem says that tan α is -1/3 and α is in the second quadrant. We need to find cos α. The options are given as A to D. Hmm, alright. Let's start by recalling some trigonometric identities and properties.First, I remember that in the second quadrant, sine is positive and cosine is negative. So whatever answer I get for cosine, it should be negative. Looking at the options, C and D are negative. So the answer is probably one of those two. Let's verify.Given tan α = -1/3. Tangent is sine over cosine, right? So tan α = sin α / cos α. Since α is in the second quadrant, sine is positive and cosine is negative. That makes sense why tangent is negative here because positive divided by negative is negative.Now, to find cosine, maybe I can use the Pythagorean identity. The identity is tan² α + 1 = sec² α. Since we know tan α, we can find sec α, and then take the reciprocal to get cos α.Let me write that down step by step.Given tan α = -1/3.First, compute tan² α: (-1/3)² = 1/9.Then, add 1 to it: 1/9 + 1 = 1/9 + 9/9 = 10/9.So sec² α = 10/9. Therefore, sec α is either sqrt(10/9) or -sqrt(10/9). But since α is in the second quadrant, cosine is negative, so sec α, which is 1/cos α, is also negative. Therefore, sec α = -sqrt(10)/3.Therefore, cos α = 1 / sec α = 1 / (-sqrt(10)/3) = -3 / sqrt(10).But we should rationalize the denominator, right? Multiply numerator and denominator by sqrt(10):-3/sqrt(10) * sqrt(10)/sqrt(10) = -3 sqrt(10) / 10.Looking at the options, D is -3 sqrt(10)/10. So that should be the answer. Let me check again.Alternatively, maybe I can think of a right triangle. Since tan α = opposite/adjacent = -1/3. But in the second quadrant, the adjacent side (which corresponds to cosine) is negative. So maybe imagine a right triangle where the opposite side is 1 (positive, since sine is positive) and the adjacent side is -3 (since cosine is negative). Then the hypotenuse would be sqrt(1² + (-3)²) = sqrt(1 + 9) = sqrt(10). Therefore, cos α = adjacent/hypotenuse = -3/sqrt(10), which is the same as before. After rationalizing, -3 sqrt(10)/10. So D.But wait, let's check the options again. A is 3√10/10, B is √10/10, C is -√10/10, D is -3√10/10. So yes, D matches. Hmm. But hold on, is there a chance I made a mistake in assigning the sides?Alternatively, maybe they consider the reference angle. Let me think. If tan α = -1/3, but since it's in the second quadrant, the reference angle would have tan θ = 1/3. Then, in the second quadrant, the coordinates would be (-3, 1) leading to hypotenuse sqrt(10). So cosine is adjacent over hypotenuse, which is -3/sqrt(10), so again -3√10/10. So yeah, D is correct.But why is option C there? C is -√10/10. Maybe if someone messed up the ratio. Maybe if they thought adjacent was 1 instead of 3? Let's see. If someone thought that tan α = -1/3, so opposite is -1, adjacent is 3, but in second quadrant, sine is positive, so opposite is 1, adjacent is -3. So hypotenuse sqrt(1 + 9) = sqrt(10). Then cosine is -3/sqrt(10), which is D. So yeah, seems correct.Alternatively, maybe if someone used sin^2 α + cos^2 α =1. Let me try that approach. Since tan α = sin α / cos α = -1/3, so sin α = (-1/3) cos α. Then substitute into sin² α + cos² α =1.So [ (-1/3 cos α )² ] + cos² α =1(1/9 cos² α) + cos² α =1Multiply through by 9 to eliminate denominators:cos² α + 9 cos² α =910 cos² α =9cos² α =9/10Therefore, cos α = sqrt(9/10) or -sqrt(9/10). But since α is in the second quadrant, cos α is negative. So cos α = -3/sqrt(10) = -3 sqrt(10)/10. Same answer. So D. So I think that's correct.Therefore, the answer should be D.Final Answerboxed{D}</think>Given (tan alpha = -frac{1}{3}) and (alpha) is an angle in the second quadrant, we need to find (cos alpha).1. Identify the signs of trigonometric functions in the second quadrant: - In the second quadrant, sine is positive and cosine is negative.2. Use the Pythagorean identity: - (tan^2 alpha + 1 = sec^2 alpha) - Calculate (tan^2 alpha): (left(-frac{1}{3}right)^2 = frac{1}{9}) - Add 1: (frac{1}{9} + 1 = frac{1}{9} + frac{9}{9} = frac{10}{9}) - Therefore, (sec^2 alpha = frac{10}{9})3. Determine (sec alpha): - Since (alpha) is in the second quadrant, (cos alpha) is negative, so (sec alpha) (which is (frac{1}{cos alpha})) is also negative. - (sec alpha = -sqrt{frac{10}{9}} = -frac{sqrt{10}}{3})4. Find (cos alpha): - (cos alpha = frac{1}{sec alpha} = frac{1}{-frac{sqrt{10}}{3}} = -frac{3}{sqrt{10}}) - Rationalize the denominator: (-frac{3}{sqrt{10}} times frac{sqrt{10}}{sqrt{10}} = -frac{3sqrt{10}}{10})5. Verify using a right triangle approach: - Consider a right triangle where the opposite side (sine) is 1 (positive) and the adjacent side (cosine) is -3 (negative). - Hypotenuse: (sqrt{1^2 + (-3)^2} = sqrt{10}) - (cos alpha = frac{-3}{sqrt{10}} = -frac{3sqrt{10}}{10})6. Conclusion: - The value of (cos alpha) is (-frac{3sqrt{10}}{10}), which corresponds to option D.(boxed{D}),<think>Okay, let me try to solve this problem. So, the question is: Given that tan α = -1/3 and α is an angle in the second quadrant, find the value of cos α. The options are A through D. Hmm, right. Let me recall my trigonometry.First, tangent is equal to sine over cosine, right? So tan α = sin α / cos α. They told us that tan α is -1/3. So that means sin α divided by cos α is -1/3. But since α is in the second quadrant, I need to remember the signs of sine and cosine there. In the second quadrant, sine is positive and cosine is negative. So sin α is positive, and cos α is negative. That might be important for determining the signs in the answer.So, we need to find cos α. Let's think about how to relate tangent to cosine. Maybe using the Pythagorean identity? Let me recall that 1 + tan² α = sec² α. Since secant is 1/cosine, so if I can find sec α, then I can find cos α.Let me write that down step by step:Given tan α = -1/3.So tan² α = (-1/3)² = 1/9.Then, 1 + tan² α = 1 + 1/9 = 10/9 = sec² α.Therefore, sec α = ±√(10/9) = ±(√10)/3.But since α is in the second quadrant, and sec α is 1/cos α, and we already established that cosine is negative in the second quadrant. So sec α, which is 1/cos α, must be negative. Therefore, sec α = -√10 / 3.Hence, cos α = 1 / sec α = 1 / (-√10 / 3) = -3 / √10.But wait, the answer options don't have that. Let me check the options again.Options are:A: 3√10 /10B: √10 /10C: -√10 /10D: -3√10 /10So, my result is -3 / √10. To rationalize the denominator, multiply numerator and denominator by √10:-3 / √10 = (-3√10) / (√10 * √10) = (-3√10)/10. So that's option D: -3√10 /10. Hmm, but wait, let me check my steps again because sometimes it's easy to mix up things.Alternatively, maybe using a right triangle approach. Since tan α = -1/3, and α is in the second quadrant. Let's imagine a right triangle where the opposite side is 1 (positive because sine is positive) and the adjacent side is -3 (since cosine is negative). Wait, but in a right triangle, lengths are positive. Maybe this approach needs adjustment.Alternatively, when dealing with angles in different quadrants, you can consider the reference angle. Let's say the reference angle is β, such that tan β = 1/3 (absolute value). Then in the second quadrant, α = π - β. So, cos α = -cos β (since cosine is negative in the second quadrant). So, if we can find cos β, then cos α is just negative of that.So, let's compute cos β where β is an acute angle with tan β = 1/3. In a right triangle with opposite side 1 and adjacent side 3, hypotenuse would be sqrt(1² + 3²) = sqrt(1 + 9) = sqrt(10). Therefore, cos β = adjacent / hypotenuse = 3 / sqrt(10). Therefore, cos α = -3 / sqrt(10) = -3√10 /10, which is option D. So that seems consistent with the previous method.Wait, but hold on, the answer options have -3√10 /10 as D. So that's the answer. But let me double-check once more.Alternatively, if I use the identity sin² α + cos² α = 1. Since tan α = sin α / cos α = -1/3, so sin α = (-1/3) cos α. Substitute into the identity:[ (-1/3 cos α )² ] + cos² α = 1(1/9 cos² α) + cos² α = 1(1/9 + 1) cos² α = 1(10/9) cos² α = 1cos² α = 9/10cos α = ± 3/√10. Now, since α is in the second quadrant, cosine is negative, so cos α = -3/√10 = -3√10 /10. Yes, same result. So that's option D. Therefore, the answer should be D.Wait a second, but looking back at the problem statement, the options are A: 3√10 /10, B: √10 /10, C: -√10 /10, D: -3√10 /10. So yes, D is the correct one. But let me check again because sometimes when using the reference angle, if I made an error in sign.But in the second quadrant, cosine is negative, so regardless of the method, it should come out negative, and the magnitude is 3/sqrt(10) which is 3√10 /10. So, combining the sign, it's -3√10 /10. Therefore, D is correct.But maybe let me check with coordinates. Suppose the angle α is in the second quadrant. Then the terminal side is in the second quadrant. The tangent is -1/3. So, if I consider a point (x, y) on the terminal side such that y/x = tan α = -1/3. But in the second quadrant, x is negative and y is positive. So, let's suppose x = -3 and y = 1 (since y/x = 1/(-3) = -1/3). Then the radius r would be sqrt(x² + y²) = sqrt(9 + 1) = sqrt(10). Therefore, cos α = x/r = -3 / sqrt(10) = -3√10 /10. Yep, same result. So definitely D. So, I think that's the answer.But wait, the options include C: -√10 /10. Maybe someone might miscalculate and get confused between sine and cosine? Let's see. If we compute sine, then sine α = y/r = 1 / sqrt(10) = √10 /10. So that would be positive, which is correct for the second quadrant. So sine is positive, cosine is negative.So, the answer is D. Therefore, I think the correct answer is D.Final Answerboxed{D}</think>Given (tan alpha = -frac{1}{3}) and (alpha) is an angle in the second quadrant, we need to find the value of (cos alpha).1. Identify the quadrant information: In the second quadrant, (sin alpha) is positive and (cos alpha) is negative.2. Use the Pythagorean identity: [ 1 + tan^2 alpha = sec^2 alpha ] Substituting (tan alpha = -frac{1}{3}): [ 1 + left(-frac{1}{3}right)^2 = sec^2 alpha implies 1 + frac{1}{9} = frac{10}{9} = sec^2 alpha ]3. Solve for (sec alpha): [ sec alpha = pm frac{sqrt{10}}{3} ] Since (alpha) is in the second quadrant, (cos alpha) is negative, so (sec alpha) (which is (frac{1}{cos alpha})) must be negative: [ sec alpha = -frac{sqrt{10}}{3} ]4. Find (cos alpha): [ cos alpha = frac{1}{sec alpha} = frac{1}{-frac{sqrt{10}}{3}} = -frac{3}{sqrt{10}} = -frac{3sqrt{10}}{10} ]5. Verify using a right triangle approach: - Consider a right triangle with opposite side 1 (positive) and adjacent side -3 (negative). - Hypotenuse: (sqrt{1^2 + (-3)^2} = sqrt{10}) - (cos alpha = frac{-3}{sqrt{10}} = -frac{3sqrt{10}}{10})Both methods confirm that (cos alpha = -frac{3sqrt{10}}{10}), which corresponds to option D.[boxed{D}]]

🔑:Since tanalpha = -frac{1}{3}, and alpha is an angle in the second quadrant,Therefore, cosalpha = -sqrt{frac{1}{1+tan^2alpha}} = -frac{3sqrt{10}}{10}.Hence, the correct option is boxed{text{D}}.By using the value of tanalpha and the fact that alpha is an angle in the second quadrant, we can find the value of cosalpha through the basic relationship between trigonometric functions of the same angle.This problem tests the application of the basic relationship between trigonometric functions of the same angle. Mastering these relationships is key to solving this problem.