Answer
$\frac{4 h^{6}}{25g^{6}}$
Work Step by Step
Given the expression: $$\left( \frac{2g^{-2}h^{-3}}{5gh^{-6}} \right)^2.$$ First we apply the quotient rule inside the parenthesis and simplify:
$$\begin{aligned}
\left( \frac{2g^{-2}h^{-3}}{5gh^{-6}} \right)^2&= \left( \frac{2 h^{-3+6}g^{-2-1}}{5} \right)^2\\\\
&= \left( \frac{2 h^{3}g^{-3}}{5} \right)^2.
\end{aligned}$$ Then we the the negative exponent rule:
$$\begin{aligned}
\left( \frac{2 h^{3}g^{-3}}{5} \right)^2=\left( \frac{2 h^{3}}{5g^3} \right)^2.
\end{aligned}$$ Finally we use the power rule:
$$\begin{aligned}
\left( \frac{2 h^{3}}{5g^3} \right)^2&= \frac{4 h^{6}}{25g^{6}}.
\end{aligned}$$
