# Chapter R - Review Exercises - Page 76: 101

$\dfrac{1}{p^2q^4}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the laws of exponents to simplify the given expression, $\dfrac{(p^{15}q^{12})^{-4/3}}{(p^{24}q^{16})^{-3/4}} .$ $\bf{\text{Solution Details:}}$ Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{p^{15\left(-\frac{4}{3} \right)}q^{12\left(-\frac{4}{3} \right)}}{p^{24\left(-\frac{3}{4} \right)}q^{16\left(-\frac{3}{4} \right)}} \\\\= \dfrac{p^{-20}q^{-16}}{p^{-18}q^{-12}} .\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} p^{-20-(-18)}q^{-16-(-12)} \\\\= p^{-20+18}q^{-16+12} \\\\= p^{-2}q^{-4} .\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{1}{p^2q^4} .\end{array}

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