Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 3 - Exponents, Polynomials and Functions - 3.1 Rules for Exponents - 3.1 Exercises: 44

Answer

$\dfrac{9}{4}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $ \left( \dfrac{2}{3} \right)^{-2} ,$ use the laws of exponents. $\bf{\text{Solution Details:}}$ Using the Power of a Quotient Rule of the laws of exponents, which is given by $\left( \dfrac{x^m}{y^n} \right)^p=\dfrac{x^{mp}}{y^{np}},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \left( \dfrac{2}{3} \right)^{-2} \\\\= \dfrac{2^{-2}}{3^{-2}} .\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{2^{-2}}{3^{-2}} \\\\= \dfrac{3^{2}}{2^{2}} \\\\= \dfrac{9}{4} .\end{array}
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