Answer
$\dfrac{b^{6}}{a^{4}}$
Work Step by Step
Using the laws of exponents, the given expression, $
\dfrac{1}{a^2b^{-3}}\left( a^2b^{-3} \right)^{-1}
,$ is equivalent to
\begin{align*}
&
\dfrac{1}{a^2b^{-3}}\left( a^{2(-1)}b^{-3(-1)} \right)
&\text{ (use $(a^x)^y=a^{xy}$)}
\\\\&=
\dfrac{1}{a^2b^{-3}}\left( a^{-2}b^{3} \right)
\\\\&=
\dfrac{a^{-2}b^{3}}{a^2b^{-3}}
\\\\&=
a^{-2-2}b^{3-(-3)}
&\left(\text{use }\dfrac{a^x}{a^y}=a^{x-y}\right)
\\\\&=
a^{-2-2}b^{3+3}
\\\\&=
a^{-4}b^{6}
\\\\&=
\dfrac{b^{6}}{a^{4}}
&\left(\text{use }a^{-x}=\dfrac{1}{a^{x}}\right)
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{b^{6}}{a^{4}}
$.
