Beginning Algebra With Applications, Chapter 7, 7.4, Section 7.4, Problem 68

Simplify: $\displaystyle \left( 2a^{-3} \right) \left( a^7 b^{-1} \right)^3$


$
\begin{equation}
\begin{aligned}

\left( 2a^{-3} \right) \left( a^7 b^{-1} \right)^3 =& (2a^{-3}) \left( a^{7(3) b^{-1(3)}} \right)
&& \text{Use the rule for Simplifying Power of Products}
\\
\\
=& (2a^{-3}) (a^{21} b^{-3})
&& \text{Simplify}
\\
\\
=& 2 \left( \frac{1}{a^3} \right) \left[ a^{21} \left( \frac{1}{b^3} \right) \right]
&& \text{Write the expression with only positive exponents}
\\
\\
=& \frac{2a^{21}}{a^3 b^3}
&& \text{Simplify}
\\
\\
=& \frac{2a^{21-3}}{b^3}
&& \text{Divide variables with the same base by subtracting the exponents}
\\
\\
=& \frac{2a^{18}}{b^3}
&&

\end{aligned}
\end{equation}
$