Answer
$\dfrac{xy}{z^2}\sqrt[6]{\dfrac{y^2}{z^3}}$
Work Step by Step
Using the properties of radicals, the given expression, $
\sqrt[6]{\dfrac{x^6y^8}{z^{15}}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[6]{x^6y^8}}{\sqrt[6]{z^{15}}}
\\\\=
\dfrac{\sqrt[6]{x^6y^6\cdot y^2}}{\sqrt[6]{z^{12}\cdot z^3}}
\\\\=
\dfrac{\sqrt[6]{(xy)^6\cdot y^2}}{\sqrt[6]{(z^{2})^6\cdot z^3}}
\\\\=
\dfrac{xy\sqrt[6]{y^2}}{z^{2}\sqrt[6]{z^3}}
\\\\=
\dfrac{xy}{z^2}\sqrt[6]{\dfrac{y^2}{z^3}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.