answer：Okay, so I need to find the range of the function ( y = log_2 (sqrt{sin x}) ) where ( x ) is between ( 0^circ ) and ( 180^circ ). Hmm, let me break this down step by step.First, I know that the sine function, ( sin x ), has a range of values between 0 and 1 for angles between ( 0^circ ) and ( 180^circ ). But since ( x ) is strictly between ( 0^circ ) and ( 180^circ ), ( sin x ) will actually be greater than 0 and less than or equal to 1. So, ( 0 < sin x leq 1 ).Next, there's a square root involved: ( sqrt{sin x} ). Taking the square root of ( sin x ) will still keep the value between 0 and 1 because the square root of a number between 0 and 1 is still between 0 and 1. So, ( 0 < sqrt{sin x} leq 1 ).Now, the function ( y = log_2 (sqrt{sin x}) ) is a logarithm with base 2. I remember that the logarithm function ( log_b a ) is defined for ( a > 0 ), which is satisfied here since ( sqrt{sin x} > 0 ). Also, since ( sqrt{sin x} leq 1 ), the logarithm will be less than or equal to 0 because ( log_2 1 = 0 ) and ( log_2 ) of a number less than 1 is negative.To find the range, I need to consider the possible values of ( sqrt{sin x} ). The smallest value ( sqrt{sin x} ) can take is just above 0, and the largest is 1. So, ( sqrt{sin x} ) is in the interval ( (0, 1] ).Applying the logarithm function to this interval, ( log_2 ) of numbers approaching 0 from the right will approach negative infinity, and ( log_2 1 = 0 ). Therefore, the range of ( y = log_2 (sqrt{sin x}) ) is all real numbers less than or equal to 0.Wait, let me double-check. If ( sqrt{sin x} ) is between 0 and 1, then ( log_2 (sqrt{sin x}) ) will indeed be between negative infinity and 0. That makes sense because as ( sqrt{sin x} ) approaches 0, the logarithm becomes more and more negative, and when ( sqrt{sin x} = 1 ), the logarithm is 0.So, putting it all together, the range of the function ( y = log_2 (sqrt{sin x}) ) for ( 0^circ < x < 180^circ ) is all real numbers less than or equal to 0, which can be written as ( (-infty, 0] ).I think that's it. I considered the domain of ( sin x ), applied the square root, and then the logarithm, making sure to account for the behavior of each function step by step. It seems logical that the range would extend to negative infinity because the logarithm can take on arbitrarily large negative values as the input approaches zero.Just to visualize, if I imagine the graph of ( y = log_2 (sqrt{sin x}) ), it would oscillate between negative infinity and 0 as ( x ) moves from just above ( 0^circ ) to just below ( 180^circ ). The maximum value of 0 occurs at ( x = 90^circ ) where ( sin x = 1 ), and it decreases towards negative infinity as ( x ) approaches ( 0^circ ) or ( 180^circ ).Yeah, I feel confident about this conclusion now.

answer：Okay, so I have this problem where I'm given that the logarithms of p, q, and (p - q) with bases 8, 12, and 18 respectively are all equal. I need to find the value of q/p. Hmm, let me think about how to approach this.First, I remember that if log_a(b) = log_c(d), then b and d are related through some exponent involving the bases a and c. Maybe I can express p, q, and (p - q) in terms of a common variable since their logarithms are equal.Let me denote the common logarithm value as k. So, I can write:log_8(p) = k log_12(q) = k log_18(p - q) = kFrom the definition of logarithms, this means:p = 8^k q = 12^k p - q = 18^kSo, substituting p and q from the first two equations into the third, I get:8^k - 12^k = 18^kHmm, that's an equation involving exponents with different bases. Maybe I can express all these bases in terms of their prime factors to simplify. Let's see:8 is 2^3, 12 is 2^2 * 3, and 18 is 2 * 3^2. So, substituting these in:(2^3)^k - (2^2 * 3)^k = (2 * 3^2)^kSimplifying the exponents:2^{3k} - (2^{2k} * 3^k) = 2^k * 3^{2k}This looks a bit complicated, but maybe I can factor out some common terms or express everything in terms of a single variable. Let me try dividing both sides by 2^{2k} * 3^k to see if that helps:(2^{3k}) / (2^{2k} * 3^k) - (2^{2k} * 3^k) / (2^{2k} * 3^k) = (2^k * 3^{2k}) / (2^{2k} * 3^k)Simplifying each term:2^{3k - 2k} / 3^k - 1 = 2^{k - 2k} * 3^{2k - k}Which simplifies to:2^k / 3^k - 1 = 2^{-k} * 3^kHmm, so now I have:(2/3)^k - 1 = (3/2)^kLet me denote x = (2/3)^k. Then, (3/2)^k is just 1/x. So substituting:x - 1 = 1/xMultiplying both sides by x to eliminate the denominator:x^2 - x = 1Bringing all terms to one side:x^2 - x - 1 = 0Now, this is a quadratic equation in terms of x. Using the quadratic formula:x = [1 ± sqrt(1 + 4)] / 2 x = [1 ± sqrt(5)] / 2Since x = (2/3)^k and both p and q are positive numbers, x must be positive. So, we take the positive root:x = [1 + sqrt(5)] / 2But wait, x is (2/3)^k, and we need to find q/p. Let's recall that:p = 8^k = (2^3)^k = 2^{3k} q = 12^k = (2^2 * 3)^k = 2^{2k} * 3^kSo, q/p = (2^{2k} * 3^k) / 2^{3k} = (3/2)^kBut earlier, we set x = (2/3)^k, so (3/2)^k = 1/x. Since x = [1 + sqrt(5)] / 2, then 1/x is 2 / [1 + sqrt(5)]. To rationalize the denominator:1/x = 2 / [1 + sqrt(5)] * [1 - sqrt(5)] / [1 - sqrt(5)] = [2(1 - sqrt(5))] / [1 - 5] = [2(1 - sqrt(5))] / (-4) = [2(sqrt(5) - 1)] / 4 = (sqrt(5) - 1) / 2So, q/p = (sqrt(5) - 1)/2. Let me check if this makes sense. The quadratic solution gave us x = [1 + sqrt(5)] / 2, which is approximately 1.618, so 1/x is approximately 0.618, which is (sqrt(5) - 1)/2. That seems reasonable.Therefore, the value of q/p is (sqrt(5) - 1)/2.**Final Answer**boxed{B}

answer：Okay, so I have this problem where a resident wants to renovate their house and they bought some wooden strips. Each strip is either 0.7 meters or 0.8 meters long. They want to know if they can combine these strips to get a specific length, and in this case, the length in question is 3.4 meters. First, I need to understand what exactly is being asked. They have two types of strips: one is 0.7 meters, and the other is 0.8 meters. By connecting some of these strips, they can make different lengths. For example, if they use two 0.7-meter strips, they get 1.4 meters. If they use one 0.7 and one 0.8, they get 1.5 meters, and so on. The question is asking which length cannot be obtained by connecting these strips, and in this case, it's 3.4 meters.So, my goal is to figure out if 3.4 meters can be made by adding up some combination of 0.7 and 0.8 meters. If it can't be made, then 3.4 meters is the answer.Let me think about how to approach this. One way is to consider all possible combinations of 0.7 and 0.8 that add up to 3.4. Since both 0.7 and 0.8 are decimal numbers, it might be a bit tricky, but I can work with them.First, let's consider how many 0.7-meter strips we might need. If we use, say, one 0.7-meter strip, then the remaining length needed would be 3.4 - 0.7 = 2.7 meters. Now, can 2.7 meters be made using 0.8-meter strips? Let's see: 0.8 * 3 = 2.4, which is less than 2.7. 0.8 * 4 = 3.2, which is more than 2.7. So, 2.7 can't be made with 0.8-meter strips. Therefore, using one 0.7-meter strip doesn't help us reach 3.4 meters.What if we use two 0.7-meter strips? Then, the total length used is 1.4 meters, and the remaining length needed is 3.4 - 1.4 = 2.0 meters. Can 2.0 meters be made with 0.8-meter strips? Let's see: 0.8 * 2 = 1.6, which is less than 2.0. 0.8 * 3 = 2.4, which is more than 2.0. So, again, 2.0 meters can't be made with 0.8-meter strips.Let's try three 0.7-meter strips. That would be 2.1 meters, and the remaining length needed is 3.4 - 2.1 = 1.3 meters. Can 1.3 meters be made with 0.8-meter strips? 0.8 * 1 = 0.8, which is less than 1.3. 0.8 * 2 = 1.6, which is more than 1.3. So, no luck there either.What about four 0.7-meter strips? That would be 2.8 meters, and the remaining length needed is 3.4 - 2.8 = 0.6 meters. Can 0.6 meters be made with 0.8-meter strips? Well, 0.8 is larger than 0.6, so we can't use any 0.8-meter strips here. So, that doesn't work.Let's try five 0.7-meter strips. That would be 3.5 meters, which is more than 3.4 meters. So, that's too much. We can't use five 0.7-meter strips.Okay, so using 0.7-meter strips doesn't seem to help us reach exactly 3.4 meters. Maybe I should try using more 0.8-meter strips and see if that works.Let's start with one 0.8-meter strip. That leaves us with 3.4 - 0.8 = 2.6 meters. Can 2.6 meters be made with 0.7-meter strips? Let's see: 0.7 * 3 = 2.1, which is less than 2.6. 0.7 * 4 = 2.8, which is more than 2.6. So, no, we can't make 2.6 meters with 0.7-meter strips.What about two 0.8-meter strips? That's 1.6 meters, leaving us with 3.4 - 1.6 = 1.8 meters. Can 1.8 meters be made with 0.7-meter strips? 0.7 * 2 = 1.4, which is less than 1.8. 0.7 * 3 = 2.1, which is more than 1.8. So, again, no luck.Three 0.8-meter strips would be 2.4 meters, leaving us with 3.4 - 2.4 = 1.0 meters. Can 1.0 meters be made with 0.7-meter strips? 0.7 * 1 = 0.7, which is less than 1.0. 0.7 * 2 = 1.4, which is more than 1.0. So, no.Four 0.8-meter strips would be 3.2 meters, leaving us with 3.4 - 3.2 = 0.2 meters. Can 0.2 meters be made with 0.7-meter strips? No, because 0.7 is larger than 0.2, and we can't use a fraction of a strip.Five 0.8-meter strips would be 4.0 meters, which is more than 3.4 meters. So, that's too much.Hmm, so using 0.8-meter strips also doesn't seem to help us reach exactly 3.4 meters.Maybe I should try combining different numbers of 0.7 and 0.8-meter strips. Let's see.Let's denote the number of 0.7-meter strips as 'a' and the number of 0.8-meter strips as 'b'. Then, the total length would be 0.7a + 0.8b = 3.4.We need to find non-negative integers a and b that satisfy this equation.Let's try to solve for a and b.First, multiply both sides by 10 to eliminate the decimals:7a + 8b = 34Now, we need to find integers a and b such that 7a + 8b = 34.Let's try different values of a and see if b comes out as an integer.Start with a = 0:7*0 + 8b = 34 => 8b = 34 => b = 34/8 = 4.25. Not an integer.a = 1:7*1 + 8b = 34 => 7 + 8b = 34 => 8b = 27 => b = 27/8 = 3.375. Not an integer.a = 2:7*2 + 8b = 34 => 14 + 8b = 34 => 8b = 20 => b = 20/8 = 2.5. Not an integer.a = 3:7*3 + 8b = 34 => 21 + 8b = 34 => 8b = 13 => b = 13/8 = 1.625. Not an integer.a = 4:7*4 + 8b = 34 => 28 + 8b = 34 => 8b = 6 => b = 6/8 = 0.75. Not an integer.a = 5:7*5 + 8b = 34 => 35 + 8b = 34 => 8b = -1. Negative b doesn't make sense.So, none of these values of a result in an integer value of b. Therefore, there are no non-negative integers a and b that satisfy the equation 7a + 8b = 34.This means that it's impossible to make a 3.4-meter strip by combining 0.7-meter and 0.8-meter strips.I think that's the conclusion. 3.4 meters cannot be obtained by connecting these wood strips.

answer：Okay, so I have this problem about a box with 100 balls, each either orange or black. The number of black balls is equally likely to be any number from 0 to 100. A black ball rolled out, and I need to find the probability that the next ball is also black. Hmm, okay, let me try to break this down.First, let me understand the setup. There are 100 balls, and each could be orange or black. The number of black balls, let's call it n, can be anywhere from 0 to 100, and each value of n is equally probable. So, the probability that there are n black balls is 1/101 for each n from 0 to 100.Now, a black ball rolled out. So, given that the first ball is black, I need to find the probability that the next ball is also black. This sounds like a conditional probability problem. I think I need to use Bayes' theorem here.Let me recall Bayes' theorem. It states that the probability of an event A given that B has occurred is equal to the probability of B given A times the probability of A, divided by the probability of B. In mathematical terms, P(A|B) = P(B|A) * P(A) / P(B).In this case, event A is "there are n black balls," and event B is "the first ball drawn is black." So, I need to find P(n | first ball is black), and then use that to find the probability that the second ball is black.Wait, actually, maybe I should think of it differently. Since the first ball is black, it affects the number of black balls left. So, if there were originally n black balls, after drawing one, there are n-1 black balls left out of 99 total balls. So, the probability that the next ball is black given that the first was black would be (n-1)/99.But since n is a random variable, I need to average this probability over all possible n, weighted by the probability that n is the actual number of black balls given that the first ball was black.So, first, I need to find the posterior distribution of n given that the first ball was black. That is, P(n | first ball is black). Using Bayes' theorem, this is equal to P(first ball is black | n) * P(n) / P(first ball is black).We know P(n) is 1/101 for each n. P(first ball is black | n) is simply n/100, since if there are n black balls out of 100, the chance of drawing a black one is n/100.Now, P(first ball is black) is the total probability over all n. So, it's the sum over n from 0 to 100 of P(first ball is black | n) * P(n). That is, sum_{n=0}^{100} (n/100) * (1/101).Let me compute that. The sum becomes (1/(100*101)) * sum_{n=0}^{100} n. The sum of n from 0 to 100 is (100*101)/2 = 5050. So, P(first ball is black) = (1/(100*101)) * 5050 = 5050 / 10100 = 1/2.Oh, interesting, so the probability that the first ball is black is 1/2. That makes sense because, on average, half the balls are black.So, now, P(n | first ball is black) = (n/100 * 1/101) / (1/2) = (n / 10100) / (1/2) = n / 5050.So, the posterior distribution of n given the first ball was black is P(n | first black) = n / 5050.Now, given this, the probability that the next ball is black is the expected value of (n-1)/99 over this posterior distribution.So, I need to compute E[(n-1)/99] where the expectation is over P(n | first black). That is, sum_{n=0}^{100} [(n-1)/99] * [n / 5050].Wait, but when n=0, (n-1)/99 is negative, which doesn't make sense. Hmm, but actually, if n=0, the probability of drawing a black ball first is zero, so in our posterior distribution, n=0 should have zero probability. Similarly, n=1 would have P(n | first black) = 1 / 5050, but (n-1)/99 would be 0. So, maybe it's okay.But let me think again. When n=0, the first ball cannot be black, so in the posterior distribution, n=0 has zero probability. Similarly, for n=1, the probability that the first ball is black is 1/100, so P(n=1 | first black) is (1/100 * 1/101) / (1/2) = (1/10100) / (1/2) = 2/10100 = 1/5050.But for n=1, the probability that the next ball is black is (1-1)/99 = 0. So, n=1 contributes zero to the expectation.Similarly, for n=2, P(n=2 | first black) = 2 / 5050, and the probability the next ball is black is (2-1)/99 = 1/99.So, in general, for each n from 1 to 100, the contribution is (n-1)/99 * (n / 5050). But actually, n starts from 1, since n=0 contributes nothing.Wait, but in the sum, n goes from 0 to 100, but n=0 contributes zero because (n-1)/99 is negative, but since P(n=0 | first black) is zero, it doesn't matter.So, the expectation is sum_{n=1}^{100} [(n-1)/99] * [n / 5050].Let me write that as (1/(99*5050)) * sum_{n=1}^{100} n(n-1).So, I need to compute sum_{n=1}^{100} n(n-1). Hmm, n(n-1) is n^2 - n, so the sum is sum_{n=1}^{100} n^2 - sum_{n=1}^{100} n.I know that sum_{n=1}^{k} n = k(k+1)/2, so for k=100, it's 100*101/2 = 5050.Similarly, sum_{n=1}^{k} n^2 = k(k+1)(2k+1)/6. For k=100, that's 100*101*201/6.Let me compute that: 100*101=10100, 10100*201=2,030,100. Then, 2,030,100 / 6 = 338,350.So, sum_{n=1}^{100} n^2 = 338,350.Therefore, sum_{n=1}^{100} n(n-1) = 338,350 - 5050 = 333,300.So, going back, the expectation is (1/(99*5050)) * 333,300.Let me compute that: 333,300 / (99*5050).First, let's compute 99*5050. 99*5000=495,000, and 99*50=4,950, so total is 495,000 + 4,950 = 499,950.So, 333,300 / 499,950.Simplify this fraction. Let's see, both numerator and denominator are divisible by 150.333,300 ÷ 150 = 2,222.499,950 ÷ 150 = 3,333.So, 2,222 / 3,333. Hmm, can this be simplified further? Let's see, 2,222 and 3,333.Divide numerator and denominator by 1,111: 2,222 ÷ 1,111 = 2, and 3,333 ÷ 1,111 = 3.So, the simplified fraction is 2/3.Therefore, the probability that the next ball is black given the first was black is 2/3.So, p = 2, q = 3, and p + q = 5.**Final Answer**The probability is boxed{5}.

answer：Alright, so I'm trying to tackle this problem about ships on a 10x10 board. A ship is defined as a figure made up of unit squares connected by common edges. The goal is to prove that if there's an odd number of different ships consisting of n unit squares, then n must be divisible by 4.First, I need to make sure I understand what constitutes a "different" ship. I think it means that two ships are different if they can't be transformed into each other by rotation or reflection. So, for example, a horizontal ship of 2 squares and a vertical ship of 2 squares are considered different because you can't rotate or reflect one to get the other.Now, the problem is about the parity of the number of ships. Specifically, if the number of different ships is odd, then n has to be divisible by 4. I need to figure out why that is the case.Maybe I should start by considering smaller values of n and see if I can spot a pattern or get some intuition. Let's see:- For n=1: There's only one ship, which is a single square. So, the number of ships is 1, which is odd. But 1 is not divisible by 4. Hmm, that seems contradictory. Maybe n=1 is a special case?- For n=2: There are two ships: one horizontal and one vertical. So, the number of ships is 2, which is even. So, n=2 doesn't satisfy the condition.- For n=3: I think there are two ships: one L-shaped and one straight. Wait, no, actually, for n=3, you can have different configurations depending on how the squares are connected. Maybe more than two? Let me think. If I place three squares in a straight line, that's one. If I place them in an L-shape, that's another. If I place them in a T-shape, is that possible? Wait, with three squares, you can't make a T-shape because the middle square would need to have three connections, but with three squares, you can only have two connections. So, maybe only two ships for n=3. So, the number is 2, which is even.- For n=4: This is where it gets interesting. There are several ships: the straight line, the square, the L-shape, and maybe some others. Let me count them. Actually, I think there are five ships for n=4. So, the number is 5, which is odd. And 4 is divisible by 4. That fits the condition.Wait, so for n=1, which is not divisible by 4, the number of ships is odd. For n=4, which is divisible by 4, the number of ships is also odd. So, maybe the condition is that if n is divisible by 4, then the number of ships is odd, but for other n, it's even? But the problem states that if the number of ships is odd, then n must be divisible by 4. So, in that case, n=1 would be an exception because it's not divisible by 4, but the number of ships is odd. Maybe n=1 is a special case because it's just a single square, which can't be rotated or reflected into anything else.But let's think about the general case. Maybe the key is in the symmetry of the board and how ships can be rotated or reflected. If a ship has certain symmetries, it might not contribute to the count in the same way as asymmetric ships.I remember that in combinatorics, when counting distinct objects under group actions (like rotations and reflections), we can use Burnside's lemma or similar tools. Maybe that's applicable here.Burnside's lemma says that the number of distinct objects is equal to the average number of fixed points of the group actions. So, in this case, the group would be the dihedral group of the square, which has 8 elements: 4 rotations (0°, 90°, 180°, 270°) and 4 reflections (over the horizontal, vertical, and the two diagonals).If I apply Burnside's lemma, the number of distinct ships would be (F0 + F90 + F180 + F270 + Fh + Fv + Fd1 + Fd2)/8, where Fθ is the number of ships fixed by the rotation θ, and Fh, Fv, Fd1, Fd2 are the numbers fixed by the respective reflections.Now, if the number of distinct ships is odd, then the total sum F0 + F90 + F180 + F270 + Fh + Fv + Fd1 + Fd2 must be divisible by 8 and result in an odd number when divided by 8. That means the total sum must be congruent to 8 mod 16, because 8 divides into it exactly once with a remainder of 0, but since the result is odd, the total sum must be 8 mod 16.But I'm not sure if this line of reasoning is correct. Maybe I need to think differently.Another approach could be to consider the parity of the number of ships. If the number of ships is odd, then the number of orbits under the group action is odd. By the orbit-stabilizer theorem, the size of each orbit divides the order of the group, which is 8. So, the size of each orbit is a divisor of 8, meaning it could be 1, 2, 4, or 8.If the number of orbits is odd, then the number of orbits of size 1 must be odd because all other orbit sizes are even. So, the number of ships fixed by all group actions (i.e., the number of ships that are symmetric under all rotations and reflections) must be odd.But what does it mean for a ship to be fixed by all group actions? It means that the ship is symmetric under all rotations and reflections, so it must be highly symmetric. For example, a square ship of size 4 is fixed by all rotations and reflections.So, if there's an odd number of such highly symmetric ships, then n must be such that these ships exist. For a ship to be fixed by all group actions, it must be centrally symmetric and have certain properties. Maybe the number of squares n must be divisible by 4 because the ship has to fit into the symmetries of the square.Wait, that makes sense. If a ship is fixed by all rotations and reflections, it must have rotational symmetry of order 4, which would require that the number of squares is divisible by 4. For example, a 4x4 square has 16 squares, which is divisible by 4. Similarly, a ship made of four squares arranged in a square is fixed by all rotations and reflections.Therefore, if the number of distinct ships is odd, there must be an odd number of such highly symmetric ships, which implies that n is divisible by 4.But I'm not entirely sure if this covers all cases. Maybe I need to think about the contribution of each orbit to the total count. If most orbits have even size, then the total number of ships would be even unless there's an odd number of orbits with size 1.So, if the number of ships is odd, there must be an odd number of orbits of size 1, which correspond to ships that are fixed by all group actions. And as we saw, these ships must have n divisible by 4.Therefore, the conclusion is that if the number of distinct ships is odd, then n must be divisible by 4.

answer：Okay, so I have this problem here: Given real numbers a and b, and the set {1, a+b, a} is equal to the set {0, b/a, b}. I need to find the value of b²⁰¹³ minus a²⁰¹³. Hmm, sets are unordered, so the elements just need to match, regardless of the order. That means each element in the first set must be in the second set and vice versa.First, let me write down the sets:First set: {1, a + b, a}Second set: {0, b/a, b}Since these sets are equal, every element in the first set must be in the second set, and every element in the second set must be in the first set. So, let's see.Looking at the first set, the elements are 1, a + b, and a. The second set has elements 0, b/a, and b.So, one of the elements in the first set must be 0 because the second set has 0. Similarly, one of the elements in the second set must be 1 because the first set has 1.So, let's start by figuring out which element in the first set is 0. The elements are 1, a + b, and a. So, either a + b = 0 or a = 0. But wait, if a = 0, then in the second set, we have b/a, which would be undefined because division by zero isn't allowed. Therefore, a cannot be 0. So, that means a + b must be 0. So, a + b = 0.Alright, so a + b = 0. That gives us a relationship between a and b: a = -b.Now, let's look at the other elements. The first set has 1, which must be equal to one of the elements in the second set. The second set has 0, b/a, and b. Since 1 is in the first set, it must be equal to either b/a or b.Similarly, the second set has b, which must be equal to one of the elements in the first set: 1, a + b, or a. But we already know a + b is 0, so b must be equal to either 1 or a.Wait, let me think step by step.From the first set, we have 1, a + b, and a. From the second set, we have 0, b/a, and b.We already established that a + b = 0, so a = -b.So, let's substitute a = -b into the sets.First set becomes: {1, 0, -b}Second set becomes: {0, b/(-b), b} which simplifies to {0, -1, b}So, now the first set is {1, 0, -b} and the second set is {0, -1, b}.Since these sets are equal, their elements must be the same. So, comparing the two sets:First set: {0, 1, -b}Second set: {0, -1, b}So, the elements must match. Therefore, the elements 1 and -1 must be in both sets.Looking at the first set, we have 1, so the second set must also have 1. But the second set has -1 and b. So, either b must be 1 or -1 must be 1, which is not possible. Therefore, b must be 1.Wait, hold on. Let me clarify.First set: {0, 1, -b}Second set: {0, -1, b}So, for these sets to be equal, every element in the first set must be in the second set, and vice versa.So, 1 must be in the second set. The second set has 0, -1, and b. So, 1 must be equal to one of these. 1 cannot be equal to 0 or -1, so 1 must be equal to b. Therefore, b = 1.Similarly, in the first set, we have -b, which is -1 because b = 1. So, the first set becomes {0, 1, -1}.The second set is {0, -1, 1}, which is the same as the first set. So, that works.So, now we have b = 1. Since a = -b, a = -1.Therefore, a = -1 and b = 1.Now, the question is to find b²⁰¹³ - a²⁰¹³.So, let's compute that.First, b²⁰¹³ is 1²⁰¹³, which is 1 because any number to any power is itself, and 1 to any power is still 1.Then, a²⁰¹³ is (-1)²⁰¹³. Now, 2013 is an odd number, so (-1) raised to an odd power is -1.Therefore, a²⁰¹³ is -1.So, b²⁰¹³ - a²⁰¹³ is 1 - (-1) = 1 + 1 = 2.Therefore, the value is 2.Wait, let me double-check to make sure I didn't make any mistakes.We started with the sets {1, a + b, a} and {0, b/a, b}. We concluded that a + b = 0, so a = -b. Then, substituting into the sets, we got {1, 0, -b} and {0, -1, b}. Then, since 1 must be in the second set, b must be 1. Therefore, a = -1.Calculating b²⁰¹³ - a²⁰¹³: 1²⁰¹³ is 1, (-1)²⁰¹³ is -1, so 1 - (-1) is 2. That seems correct.I think that's solid. So, the answer should be 2.**Final Answer**The value of ( b^{2013} - a^{2013} ) is boxed{2}.