Answer
$3x^{3}y^{4}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[4]{81x^{12}y^{16}}
,$ find a factor of the radicand that is a perfect power of the index. Then extract the root of that factor. Note that all variables are assumed to represent positive real numbers.
$\bf{\text{Solution Details:}}$
Expressing the radicand of the expression above with a factor that is a perfect power of the index and then extracting the root of that factor results to
\begin{array}{l}\require{cancel}
\sqrt[4]{(3x^{3}y^{4})^4}
\\\\=
3x^{3}y^{4}
.\end{array}