#### Answer

$\dfrac{3a}{b^2}\sqrt[5]{\dfrac{a^4}{b^3}}$

#### Work Step by Step

Using the properties of radicals, the given expression, $
\sqrt[5]{\dfrac{243a^9}{b^{13}}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[5]{243a^9}}{\sqrt[5]{b^{13}}}
\\\\=
\dfrac{\sqrt[5]{243a^5\cdot a^4}}{\sqrt[5]{b^{10}\cdot b^3}}
\\\\=
\dfrac{\sqrt[5]{(3a)^5\cdot a^4}}{\sqrt[5]{(b^{2})^5\cdot b^3}}
\\\\=
\dfrac{3a\sqrt[5]{a^4}}{b^{2}\sqrt[5]{b^3}}
\\\\=
\dfrac{3a}{b^2}\sqrt[5]{\dfrac{a^4}{b^3}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.