Answer
$81x^2y^4$
Work Step by Step
Recall:
(1) $a^{\frac{m}{n}}=\sqrt[n]{a^m}$
(2) $\left(a^m\right)^n=a^{mn}$
(3) $(ab)^m=a^mb^m$
Use rule (1) above to obtain:
\begin{align*}
\left(\sqrt{9xy^2}\right)^4&=\left(\left(9xy^2\right)^{\frac{1}{2}}\right)^4\\
\end{align*}
Use rule (2) above to obtain:
\begin{align*}
\left(\left(9xy^2\right)^{\frac{1}{2}}\right)^4&=\left(9xy^2\right)^{\frac{1}{2} \cdot 4}\\
&=(9xy^2)^2
\end{align*}
Use rule (3) above to obtain:
\begin{align*}
(9xy^2)^2&=9^2x^2(y^2)^2\\
&=81x^2y^4
\end{align*}
