Answer
$9c^{24}d^{32}$
Work Step by Step
Extracting the factors that are perfect powers of the index, the given expression, $
\sqrt{81c^{48}d^{64}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
\sqrt{(9c^{24}d^{32})^2}
\\\\&=
9c^{24}d^{32}
.\end{align*}
Hence, the given expression simplifies to $
9c^{24}d^{32}
$.