You're on a roll — Step 1: Express $\sqrt[3]{xy^{-3}}$ as a positive power of $xy$. First, rewrite the cube root as a fractional exponent: $$\sqrt[3]{xy^{-3}} = (xy^{-3})^{\frac{1}{3}}$$ Apply the exponent $\frac{1}{3}$ to each term inside the parentheses: $$(x)^{\frac{1}{3}} (y^{-3})^{\frac{1}{3}}$$ Simplify the exponents: $$x^{\frac{1}{3}} y^{-3 \times \frac{1}{3}}$$ $$x^{\frac{1}{3}} y^{-1}$$ To express this as a positive power of $xy$, we need to handle the $y^{-1}$ term. Recall that $y^{-1} = \frac{1}{y}$. So, the expression becomes: $$x^{\frac{1}{3}} \cdot \frac{1}{y} = \frac{x^{\frac{1}{3}}}{y}$$ 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: $$(xy^{-3})^{\frac{1}{3}}$$ 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 $x^{\frac{1}{3}} y^{-1}$ to have a base of $xy$. We can write $y^{-1}$ as $(y^3)^{-\frac{1}{3}}$. 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. $$(xy^{-3})^{\frac{1}{3}} = x^{\frac{1}{3}} y^{-1}$$ To make the power of $y$ positive, we move it to the denominator: $$x^{\frac{1}{3}} y^{-1} = \frac{x^{\frac{1}{3}}}{y}$$ 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 $x^{\frac{1}{3}} y^{-1}$. We know $(xy)^k = x^k y^k$. If $k = \frac{1}{3}$, then $(xy)^{\frac{1}{3}} = x^{\frac{1}{3}} y^{\frac{1}{3}}$. This is not $x^{\frac{1}{3}} y^{-1}$. If $k = -1$, then $(xy)^{-1} = x^{-1} y^{-1}$. This is not $x^{\frac{1}{3}} y^{-1}$. 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 $(xy^{-3})^{\frac{1}{3}}$. The exponent $\frac{1}{3}$ is positive. The base is $xy^{-3}$. If the question implies the base must be $xy$, then $x^{\frac{1}{3}}y^{-1}$ 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: $$\frac{x^{\frac{1}{3}}}{y}$$ This is the most simplified form with positive exponents. It is not a single power of $xy$. Let's re-read carefully: "Express $\sqrt[3]{xy^{-3}}$ 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 $\sqrt[3]{(xy)^{-3}}$. If it was $\sqrt[3]{(xy)^{-3}}$: $$((xy)^{-3})^{\frac{1}{3}} = (xy)^{-3 \times \frac{1}{3}} = (xy)^{-1}$$ This is not a positive power. Let's assume it meant $\sqrt[3]{(xy)^3}$. If it was $\sqrt[3]{(xy)^3}$: $$((xy)^3)^{\frac{1}{3}} = (xy)^{3 \times \frac{1}{3}} = (xy)^1$$ This is a positive power of $xy$. But the question clearly shows $xy^{-3}$. Given the exact expression $\sqrt[3]{xy^{-3}}$: $$(xy^{-3})^{\frac{1}{3}} = x^{\frac{1}{3}} y^{-1}$$ It is not possible to write $x^{\frac{1}{3}} y^{-1}$ 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 $(xy^{-3})^{\frac{1}{3}}$ has a positive power $\frac{1}{3}$. 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. $x^{\frac{1}{3}} y^{-1} = \frac{x^{\frac{1}{3}}}{y^1}$. 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, $(xy^{-3})^{\frac{1}{3}}$ is already a positive power of $xy^{-3}$. If the question means to express it as $(xy)^k$ where $k>0$: Let $x^{\frac{1}{3}} y^{-1} = (xy)^k = x^k y^k$. Then $k = \frac{1}{3}$ for $x$, and $k = -1$ for $y$. This is a contradiction. Therefore, it is not possible to express $\sqrt[3]{xy^{-3}}$ 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 $\sqrt[3]{x^3y^{-3}}$ as a positive power of $xy$": $$\sqrt[3]{x^3y^{-3}} = (x^3y^{-3})^{\frac{1}{3}} = x^{3 \times \frac{1}{3}} y^{-3 \times \frac{1}{3}} = x^1 y^{-1} = \frac{x}{y}$$ This is still not a positive power of $xy$. Let's assume the question meant "Express $\sqrt[3]{x^3y^3}$ as a positive power of $xy$": $$\sqrt[3]{x^3y^3} = (x^3y^3)^{\frac{1}{3}} = x^{3 \times \frac{1}{3}} y^{3 \times \frac{1}{3}} = x^1 y^1 = xy = (xy)^1$$ 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, $(xy^{-3})^{\frac{1}{3}}$ 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 $x^{\frac{1}{3}}y^{-1}$ 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 $\sqrt[3]{xy^{-3}}$ with positive exponents." $$\sqrt[3]{xy^{-3}} = (xy^{-3})^{\frac{1}{3}} = x^{\frac{1}{3}} y^{-1} = \frac{x^{\frac{1}{3}}}{y}$$ 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 $k>0$: $x^{\frac{1}{3}}y^{-1} = (xy)^k \implies x^{\frac{1}{3}}y^{-1} = x^k y^k$. This implies $k = \frac{1}{3}$ 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 $\sqrt[3]{xy^{-3}}$. 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 $\sqrt[3]{xy^{-3}}$ using positive exponents." Step 1: Rewrite the cube root as a fractional exponent. $$\sqrt[3]{xy^{-3}} = (xy^{-3})^{\frac{1}{3}}$$ Step 2: Apply the exponent to each term inside the parentheses. $$x^{\frac{1}{3}} (y^{-3})^{\frac{1}{3}} = x^{\frac{1}{3}} y^{-3 \times \frac{1}{3}} = x^{\frac{1}{3}} y^{-1}$$ Step 3: Rewrite with positive exponents. $$x^{\frac{1}{3}} y^{-1} = \frac{x^{\frac{1}{3}}}{y}$$ 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 $x^{\frac{1}{3}}y^{-1}$ 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 $(xy)^k$. $x^{\frac{1}{3}}y^{-1} = x^{\frac{1}{3}}y^{\frac{1}{3}}y^{-\frac{1}{3}}y^{-1} = (xy)^{\frac{1}{3}} y^{-\frac{4}{3}}$. 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 $\sqrt[3]{xy^{-3}}$ as a positive power of $xy$. Working out: $$\sqrt[3]{xy^{-3}} = (xy^{-3})^{\frac{1}{3}}$$ $$= x^{\frac{1}{3}} (y^{-3})^{\frac{1}{3}}$$ $$= x^{\frac{1}{3}} y^{-1}$$ To express this with positive powers, we write $y^{-1}$ as $\frac{1}{y}$: $$= \frac{x^{\frac{1}{3}}}{y}$$ 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 $x^{\frac{1}{3}}$ is a positive power, and $y$ in the denominator is also a positive power. I will provide the simplified form. $$\boxed{\frac{x^{\frac{1}{3}}}{y}}$$ Step 2: Evaluate: $\sqrt[3]{8^2}$. Working out: First, calculate $8^2$: $$8^2 = 8 \times 8 = 64$$ Now, find the cube root of 64: $$\sqrt[3]{64}$$ We need to find a number that, when multiplied by itself three times, equals 64. $$4 \times 4 \times 4 = 16 \times 4 = 64$$ So, $\sqrt[3]{64} = 4$. Alternatively, using fractional exponents: $$\sqrt[3]{8^2} = (8^2)^{\frac{1}{3}} = 8^{\frac{2}{3}}$$ Since $8 = 2^3$: $$(2^3)^{\frac{2}{3}} = 2^{3 \times \frac{2}{3}} = 2^2$$ $$2^2 = 4$$ $$\boxed{4}$$ 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: $$200 = 100 \times 2$$ Now, apply the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$: $$\sqrt{200} = \sqrt{100 \times 2}$$ $$= \sqrt{100} \times \sqrt{2}$$ Calculate the square root of 100: $$\sqrt{100} = 10$$ Substitute this back into the expression: $$10 \times \sqrt{2} = 10\sqrt{2}$$ $$\boxed{10\sqrt{2}}$$ Got more? Send 'em!