Answer
$\dfrac{ab^2}{c^2}\sqrt[6]{\dfrac{a^3}{c}}$
Work Step by Step
Using the properties of radicals, the given expression, $
\sqrt[6]{\dfrac{a^9b^{12}}{c^{13}}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[6]{a^9b^{12}}}{\sqrt[6]{c^{13}}}
\\\\=
\dfrac{\sqrt[6]{a^6b^{12}\cdot a^3}}{\sqrt[6]{c^{12}\cdot c}}
\\\\=
\dfrac{\sqrt[6]{(ab^{2})^6\cdot a^3}}{\sqrt[6]{(c^{2})^6\cdot c}}
\\\\=
\dfrac{ab^{2}\sqrt[6]{a^3}}{c^{2}\sqrt[6]{c}}
\\\\=
\dfrac{ab^2}{c^2}\sqrt[6]{\dfrac{a^3}{c}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.