#### Answer

$\dfrac{a^{2}}{2}$

#### Work Step by Step

Using the properties of radicals, the expression, $
\sqrt[5]{\dfrac{3a^{12}}{96a^2}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[5]{\dfrac{\cancel3a^{12-2}}{\cancel3\cdot32}}
\\\\=
\sqrt[5]{\dfrac{a^{10}}{32}}
\\\\=
\sqrt[5]{\left( \dfrac{a^{2}}{2} \right)^5}
\\\\=
\dfrac{a^{2}}{2}
.\end{array}
Note that the variable/s is/are assumed to be positive.