College Algebra, Exercise P, Exercise P.4, Section Exercise P.4, Problem 52

Simplify the expression $\displaystyle \left( 2a^3 b^2 \right)^2 \left( 5a^2b^5 \right)^3$ and eliminate any negative exponents.

$
\begin{equation}
\begin{aligned}
\left( 2a^3 b^2 \right)^2 \left( 5a^2b^5 \right)^3 &= \left[ 2^2(a^3)^2(b^2)^2 \right] \left[ 5^3(a^2)^3(b^5)^3 \right] && \text{Law: } (ab)^n = a^n b^n\\
\\
&= \left( 4a^6 b^4 \right) \left(125a^6 b^{15} \right) && \text{Law: } (a^m)^n = a^{mn}\\
\\
&= (4)(125)a^6 a^6 b^4 b^{15} && \text{Group factors with same base}\\
\\
&= 500 a^{6+6} b^{4+15} && \text{Law: } a^m a^n = a^{m+n}\\
\\
&= 500 a^{12} b^{19}
\end{aligned}
\end{equation}
$