The problem asks us to simplify the expression $(\frac{8x^4}{y})^{-2}$ and express the answer with positive exponents.

1. Problem Description

The problem asks us to simplify the expression $(\frac{8x^4}{y})^{-2}$ and express the answer with positive exponents.

2. Solution Steps

We have the expression $(\frac{8x^4}{y})^{-2}$.

First, we apply the rule $(a/b)^n = a^n / b^n$.

$(\frac{8x^4}{y})^{-2} = \frac{(8x^4)^{-2}}{y^{-2}}$

Next, we apply the rule $(ab)^n = a^n b^n$.

$\frac{(8x^4)^{-2}}{y^{-2}} = \frac{8^{-2}(x^4)^{-2}}{y^{-2}}$

Next, we apply the rule $(a^m)^n = a^{mn}$.

$\frac{8^{-2}(x^4)^{-2}}{y^{-2}} = \frac{8^{-2}x^{-8}}{y^{-2}}$

Now, we use the rule $a^{-n} = \frac{1}{a^n}$

$\frac{8^{-2}x^{-8}}{y^{-2}} = \frac{\frac{1}{8^2}\frac{1}{x^8}}{\frac{1}{y^2}}$

$\frac{\frac{1}{8^2}\frac{1}{x^8}}{\frac{1}{y^2}} = \frac{1}{8^2 x^8} \cdot \frac{y^2}{1} = \frac{y^2}{8^2 x^8}$

Finally, we calculate $8^2 = 64$.

$\frac{y^2}{8^2 x^8} = \frac{y^2}{64x^8}$

3. Final Answer

The simplified expression is $\frac{y^2}{64x^8}$.