Answer
$\dfrac{2a}{bc^2}$
Work Step by Step
Using the properties of radicals, the given expression, $
\sqrt[4]{\dfrac{32a^4}{2b^4c^8}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[4]{\dfrac{16a^4}{b^4c^8}}
\\\\=
\dfrac{\sqrt[4]{16a^4}}{\sqrt[4]{b^4c^8}}
\\\\=
\dfrac{\sqrt[4]{(2a)^4}}{\sqrt[4]{(bc^2)^4}}
\\\\=
\dfrac{2a}{bc^2}
\end{array}
* Note that it is assumed that all variables represent positive numbers.