Answer
$\text{The given expression simplifies to }\dfrac{1}{16}$.
Work Step by Step
$\begin{array}{ l c }
=\dfrac{1}{( -32)^{\dfrac{4}{5}}} & \begin{array}{l}
\mathrm{Apply\ the\ rule\ }\\
a^{-b} =\dfrac{1}{a^{b}}
\end{array}\\
& \\
=\dfrac{1}{\left(( -32)^{\dfrac{1}{5}}\right)^{4}} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
a^{n\cdot m} =\left( a^{n}\right)^{m}\\
\end{array}\\
& \\
=\dfrac{1}{\left(\sqrt[5]{-32}\right)^{4}} & \begin{array}{l}
\mathrm{Apply\ the\ rule}\\
a^{\frac{b}{c}} =\sqrt[c]{a^{b}}
\end{array}\\
& \\
=\dfrac{1}{( -2)^{4}} & \mathrm{Simplify}\\
& \\
=\dfrac{1}{16} &
\end{array}$
