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

1. Problem Description

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

2. Solution Steps

We are given the expression $(\frac{5x^2}{y})^{-2}$.

First, we can use the property that $(\frac{a}{b})^n = \frac{a^n}{b^n}$. So,

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

Next, we can use the property that $(ab)^n = a^n b^n$. So,

$(5x^2)^{-2} = 5^{-2}(x^2)^{-2}$.

Also, we can use the property that $(a^m)^n = a^{mn}$. So,

$(x^2)^{-2} = x^{2*(-2)} = x^{-4}$.

Therefore, $(5x^2)^{-2} = 5^{-2} x^{-4}$.

Substituting this back into our expression, we have:

$\frac{(5x^2)^{-2}}{y^{-2}} = \frac{5^{-2}x^{-4}}{y^{-2}}$.

To express with positive exponents, we can use the property that $a^{-n} = \frac{1}{a^n}$. So, $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$, $x^{-4} = \frac{1}{x^4}$, and $y^{-2} = \frac{1}{y^2}$. Therefore, $\frac{1}{y^{-2}} = y^2$.

So our expression becomes:

$\frac{5^{-2}x^{-4}}{y^{-2}} = \frac{\frac{1}{5^2} \cdot \frac{1}{x^4}}{\frac{1}{y^2}} = \frac{\frac{1}{25x^4}}{\frac{1}{y^2}} = \frac{1}{25x^4} \cdot \frac{y^2}{1} = \frac{y^2}{25x^4}$.

3. Final Answer

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