#### Answer

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

#### Work Step by Step

Using the properties of radicals, the given expression, $
\sqrt[3]{\dfrac{27a^4}{8b^3}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[3]{27a^4}}{\sqrt[3]{8b^3}}
\\\\=
\dfrac{\sqrt[3]{27a^3\cdot a}}{\sqrt[3]{8b^3}}
\\\\=
\dfrac{\sqrt[3]{(3a)^3\cdot a}}{\sqrt[3]{(2b)^3}}
\\\\=
\dfrac{3a\sqrt[3]{a}}{2b}
\end{array}
* Note that it is assumed that all variables represent positive numbers.