College Algebra, Exercise P, Exercise P.4, Section Exercise P.4, Problem 72

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

$\begin{equation} \begin{aligned} \left( \frac{xy^{-2}z^{-3}}{x^2y^3z^{-4}} \right)^{-3} &= \left( \frac{x^2y^3 z^{-4}}{xy^{-2}z^{-3}} \right)^{3} && \text{Law: } \left( \frac{a}{b} \right)^{-n} = \left( \frac{b}{a} \right)^n\\ \\ &= \frac{(x^2)^3(y^3)^3(z^{-4})^3}{x^3(y^{-2})^3(z^{-3})^3} && \text{Law: } (ab)^n = a^nb^n\\ \\ &= \frac{x^6 y^9 z^{-12}}{x^3y^{-6}z^{-9}} && \text{Law: } \frac{a^m}{a^n} = a^{m-n}\\ \\ &= x^{6-3} y^{9-(-6)} z^{-12-(-9)} && \text{Simplify}\\ \\ &= x^3 y^{15} z^{-3} && \text{Definition of negative exponent } a^{-n} = \frac{1}{a^n}\\ \\ &= \frac{x^3y^{15}}{z^3} \end{aligned} \end{equation}$