The problem asks to simplify the expression $(\frac{5xy^{-2}}{z^2})^2$ and express the answer with positive exponents.

1. Problem Description

The problem asks to simplify the expression $(\frac{5xy^{-2}}{z^2})^2$ and express the answer with positive exponents.

2. Solution Steps

First, apply the power of a quotient rule, which states that $(\frac{a}{b})^n = \frac{a^n}{b^n}$.

$$(\frac{5xy^{-2}}{z^2})^2 = \frac{(5xy^{-2})^2}{(z^2)^2}$$

Next, apply the power of a product rule, which states that $(ab)^n = a^n b^n$.

$$\frac{(5xy^{-2})^2}{(z^2)^2} = \frac{5^2 x^2 (y^{-2})^2}{(z^2)^2}$$

Simplify the exponents. Also $5^2 = 25$.

$$\frac{25x^2 y^{-4}}{z^4}$$

To express the answer with positive exponents, we rewrite $y^{-4}$ as $\frac{1}{y^4}$.

$$\frac{25x^2}{y^4 z^4}$$

3. Final Answer

$\frac{25x^2}{y^4z^4}$