College Algebra, Exercise P, Exercise P.4, Section Exercise P.4, Problem 62

Simplify the expression $\displaystyle \left( \frac{x^4z^2}{4y^5} \right)\left( \frac{2xz^3y^2}{z^3} \right)^2$ and eliminate any negative exponents.

$
\begin{equation}
\begin{aligned}
\left( \frac{x^4z^2}{4y^5} \right)\left( \frac{2xz^3y^2}{z^3} \right)^2 &= \left( \frac{x^4z^2}{4y^5} \right) \left[ \frac{2(x^3)^2(y^2)^2}{(z^3)^2} \right] && \text{Law: } (ab)^n = a^nb^n\\
\\
&= \left( \frac{x^4z^2}{4y^5} \right) \left( \frac{2x^6y^4}{z^6} \right) && \text{Law: } (a^m)^n = a^{mn}\\
\\
&= \frac{2x^{4+6}y^{4-5}z^{2-6}}{4} && \text{Law: } a^m a^n = a^{m+n}\\
\\
&= \frac{x^{10}y^{-1}z^{-4}}{2} && \text{Definition of negative exponent } a^{-n} = \frac{1}{a^n}\\
\\
&= \frac{x^{10}}{2yz^4}
\end{aligned}
\end{equation}
$