Answer
$\dfrac{y\sqrt[4]{xy^2}}{3}$
Work Step by Step
Using the properties of radicals, the given expression, $
\sqrt[4]{\dfrac{xy^6}{81}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[4]{\dfrac{y^4}{81}\cdot xy^2}
\\\\=
\sqrt[4]{\left( \dfrac{y}{3}\right)^4\cdot xy^2}
\\\\=
\dfrac{y}{3}\sqrt[4]{xy^2}
\\\\=
\dfrac{y\sqrt[4]{xy^2}}{3}
.\end{array}
Note that all variables are assumed to have positive real numbers.