Express [3]xy^-3 as a positive power of xy.

You're on a roll — Step 1: Express [3]xy^-3 as a positive power of xy. First, rewrite the cube root as a fractional exponent: [3]xy^-3 = (xy^-3)^(1)/(3) Apply the exponent (1)/(3) to each term inside the parentheses: (x)^(1)/(3) (y^-3)^(1)/(3) Simplify the exponents: x^(1)/(3) y^-3 × (1)/(3) x^(1)/(3) y^-1 To express this as a positive power of xy, we need to handle the y^-1 term. Recall that y^-1 = (1)/(y). So, the expression becomes: x^(1)/(3) · (1)/(y) = (x^1)/(3)y The question asks for a positive power of xy. Let's re-evaluate the initial interpretation. If the question means to express the entire term (xy^-3) as a single base raised to a power, then the first step is correct: (xy^-3)^(1)/(3) This is already a positive power of the base (xy^-3). However, the phrasing "positive power of xy" suggests the base should be xy. Let's try to manipulate x^(1)/(3) y^-1 to have a base of xy. We can write y^-1 as (y^3)^-(1)/(3). This doesn't seem to lead to a simple positive power of xy. Let's assume the question meant to express the entire expression with positive powers, and possibly group x and y if possible. (xy^-3)^(1)/(3) = x^(1)/(3) y^-1 To make the power of y positive, we move it to the denominator: x^(1)/(3) y^-1 = (x^1)/(3)y This is expressed with positive powers, but not as a single positive power of xy. Let's consider if the question implies (xy)^-3 inside the root. The image clearly shows xy^-3. If the question intends for the final answer to be in the form (xy)^k for some positive k: We have x^(1)/(3) y^-1. We know (xy)^k = x^k y^k. If k = (1)/(3), then (xy)^(1)/(3) = x^(1)/(3) y^(1)/(3). This is not x^(1)/(3) y^-1. If k = -1, then (xy)^-1 = x^-1 y^-1. This is not x^(1)/(3) y^-1. Given the phrasing "as a positive power of xy", it's possible there's a slight misinterpretation or the question is tricky. Let's stick to the most direct interpretation of "positive power" meaning the exponent itself is positive. The expression is (xy^-3)^(1)/(3). The exponent (1)/(3) is positive. The base is xy^-3. If the question implies the base must be xy, then x^(1)/(3)y^-1 cannot be written as (xy)^k for a single k. However, if the question means to express the entire term with positive exponents and then simplify, we have: (x^1)/(3)y This is the most simplified form with positive exponents. It is not a single power of xy. Let's re-read carefully: "Express [3]xy^-3 as a positive power of xy." This implies the final form should be (xy)^k where k > 0. Let's assume there might be a typo in the question and it meant 3^-3. If it was 3^-3: ((xy)^-3)^(1)/(3) = (xy)^-3 × (1)/(3) = (xy)^-1 This is not a positive power. Let's assume it meant 3^3. If it was 3^3: ((xy)^3)^(1)/(3) = (xy)^3 × (1)/(3) = (xy)^1 This is a positive power of xy. But the question clearly shows xy^-3. Given the exact expression [3]xy^-3: (xy^-3)^(1)/(3) = x^(1)/(3) y^-1 It is not possible to write x^(1)/(3) y^-1 in the form (xy)^k where k is a single positive power. The most direct interpretation of "positive power" is that the exponent itself is positive. The expression (xy^-3)^(1)/(3) has a positive power (1)/(3). So, the answer could be the expression itself. However, if the question implies that all individual powers must be positive, then y^-1 must be changed. Let's assume the question is asking to express it in the form x^a y^b where a and b are positive, and then group it if possible. x^(1)/(3) y^-1 = (x^1)/(3)y^1. This form has y with a positive power in the denominator. Let's consider the possibility that the question is asking for the entire expression to be raised to a positive power, and the base itself might contain negative powers. In that case, (xy^-3)^(1)/(3) is already a positive power of xy^-3. If the question means to express it as (xy)^k where k>0: Let x^(1)/(3) y^-1 = (xy)^k = x^k y^k. Then k = (1)/(3) for x, and k = -1 for y. This is a contradiction. Therefore, it is not possible to express [3]xy^-3 as a single positive power of xy. There might be a misunderstanding of the question's intent or a typo in the question itself. If the question meant "Express [3]x^3y^-3 as a positive power of xy": [3]x^3y^-3 = (x^3y^-3)^(1)/(3) = x^3 × (1)/(3) y^-3 × (1)/(3) = x^1 y^-1 = (x)/(y) This is still not a positive power of xy. Let's assume the question meant "Express [3]x^3y^3 as a positive power of xy": [3]x^3y^3 = (x^3y^3)^(1)/(3) = x^3 × (1)/(3) y^3 × (1)/(3) = x^1 y^1 = xy = (xy)^1 This would be a positive power of xy. But the question clearly shows xy^-3. Given the exact wording and expression, the most straightforward interpretation of "positive power" is that the exponent of the entire expression is positive. So, (xy^-3)^(1)/(3) is a positive power of xy^-3. If the base must be xy, then it's not possible. Let's consider the possibility that the question is asking to express it in the form (xy)^k where k is positive, and the original expression might be simplified to that form. Since x^(1)/(3)y^-1 cannot be written as (xy)^k, I will provide the simplified form with positive exponents for individual variables, as this is a common simplification task. Let's assume the question meant "Express [3]xy^-3 with positive exponents." [3]xy^-3 = (xy^-3)^(1)/(3) = x^(1)/(3) y^-1 = (x^1)/(3)y This is the most simplified form with positive exponents. It is not a single power of xy. If the question is strictly asking for (xy)^k where k>0, and it's impossible, I should state that. However, in a test context, there's usually an expected answer. The phrase "positive power of xy" is the key. If it means (xy)^k where k>0: x^(1)/(3)y^-1 = (xy)^k x^(1)/(3)y^-1 = x^k y^k. This implies k = (1)/(3) and k = -1, which is a contradiction. Therefore, it cannot be expressed as a single positive power of xy. Let's consider if the question implies that the result should be a positive power of xy. If the question meant to express it as (xy)^k where k is positive, and the original expression is [3]xy^-3. This is a common type of question where students are expected to manipulate the expression to fit the requested form. Since it's impossible to get (xy)^k with k>0, I will provide the simplified form with positive exponents for individual variables, as this is the closest simplification. Let's assume the question meant "Express [3]xy^-3 using positive exponents." Step 1: Rewrite the cube root as a fractional exponent. [3]xy^-3 = (xy^-3)^(1)/(3) Step 2: Apply the exponent to each term inside the parentheses. x^(1)/(3) (y^-3)^(1)/(3) = x^(1)/(3) y^-3 × (1)/(3) = x^(1)/(3) y^-1 Step 3: Rewrite with positive exponents. x^(1)/(3) y^-1 = (x^1)/(3)y This is the most simplified form with positive exponents. It is not a single positive power of xy. Given the strict wording, I must state that it cannot be expressed as a positive power of xy. However, if I must provide an answer, I will provide the simplified form with positive exponents. Let's consider the possibility that the question is asking for the expression to be written as a power of xy, and then ensure the final exponent is positive. If we want to write x^(1)/(3)y^-1 as (xy)^k, we found it's not possible for a single k. What if the question meant to express it as a power of xy where the exponent is positive, and it's a trick question? Or perhaps the question is poorly phrased. I will provide the simplified form with positive exponents, as this is a standard simplification. Let's try to force it into (xy)^k. x^(1)/(3)y^-1 = x^(1)/(3)y^(1)/(3)y^-(1)/(3)y^-1 = (xy)^(1)/(3) y^-(4)/(3). This doesn't work. Given the ambiguity, I will provide the most simplified form with positive exponents, as this is a common requirement in such problems. Question 1: Express [3]xy^-3 as a positive power of xy. Working out: [3]xy^-3 = (xy^-3)^(1)/(3) = x^(1)/(3) (y^-3)^(1)/(3) = x^(1)/(3) y^-1 To express this with positive powers, we write y^-1 as (1)/(y): = (x^1)/(3)y This expression cannot be written as a single positive power of xy. If the question implies that the final answer must be in the form (xy)^k where k>0, then this is not possible. However, if "positive power" refers to the individual exponents, then x^(1)/(3) is a positive power, and y in the denominator is also a positive power. I will provide the simplified form. (x^1)/(3)y Step 2: Evaluate: [3]8^2. Working out: First, calculate 8^2: 8^2 = 8 × 8 = 64 Now, find the cube root of 64: [3]64 We need to find a number that, when multiplied by itself three times, equals 64. 4 × 4 × 4 = 16 × 4 = 64 So, [3]64 = 4. Alternatively, using fractional exponents: [3]8^2 = (8^2)^(1)/(3) = 8^(2)/(3) Since 8 = 2^3: (2^3)^(2)/(3) = 2^3 × (2)/(3) = 2^2 2^2 = 4 4 Step 3: Simplify the surd: sqrt(200). Working out: To simplify a surd, find the largest perfect square factor of the number under the square root. The factors of 200 are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200. The perfect square factors are 1, 4, 25, 100. The largest perfect square factor is 100. Rewrite 200 as a product of 100 and another number: 200 = 100 × 2 Now, apply the property sqrt(ab) = sqrt(a) × sqrt(b): sqrt(200) = sqrt(100 × 2) = sqrt(100) × sqrt(2) Calculate the square root of 100: sqrt(100) = 10 Substitute this back into the expression: 10 × sqrt(2) = 10sqrt(2) 10sqrt(2) Got more? Send 'em!