Answer
$\dfrac{4}{a^2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\dfrac{4a^5(a^{-1})^3}{(a^{-2})^{-2}}
.$
$\bf{\text{Solution Details:}}$
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4a^5a^{-1(3)}}{a^{-2(-2)}}
\\\\=
\dfrac{4a^5a^{-3}}{a^{4}}
.\end{array}
Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4a^{5+(-3)}}{a^{4}}
\\\\=
\dfrac{4a^{5-3}}{a^{4}}
\\\\=
\dfrac{4a^{2}}{a^{4}}
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
4a^{2-4}
\\\\=
4a^{-2}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4}{a^2}
.\end{array}
