Answer
$\dfrac{a^{12}}{16b^{20}}$
Work Step by Step
Using $\left(\dfrac{a}{b}\right)^x=\dfrac{a^x}{b^x}$, then the expression, $
\left(\dfrac{a^3}{-2b^5}\right)^4
$, simplifies to
\begin{array}{l}\require{cancel}
\dfrac{(a^3)^4}{(-2b^5)^4}
.\end{array}
Using $(a^x)^y=a^{xy}$, then the given expression, $
\dfrac{(a^3)^4}{(-2b^5)^4}
$, simplifies to
\begin{array}{l}\require{cancel}
\dfrac{a^{3(4)}}{(-2)^4b^{5(4)}}
\\\\=
\dfrac{a^{12}}{16b^{20}}
.\end{array}