Answer
$\dfrac{ab^2}{c^{2}}\sqrt[4]{\dfrac{a}{c^2}}$
Work Step by Step
Using the properties of radicals, the given expression, $
\sqrt[4]{\dfrac{a^5b^8}{c^{10}}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[4]{a^5b^8}}{\sqrt[4]{c^{10}}}
\\\\=
\dfrac{\sqrt[4]{a^4b^8\cdot a}}{\sqrt[4]{c^{8}\cdot c^2}}
\\\\=
\dfrac{\sqrt[4]{(ab^2)^4\cdot a}}{\sqrt[4]{(c^{2})^4\cdot c^2}}
\\\\=
\dfrac{ab^2\sqrt[4]{a}}{c^{2}\sqrt[4]{c^2}}
\\\\=
\dfrac{ab^2}{c^{2}}\sqrt[4]{\dfrac{a}{c^2}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.