Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 7 - Section 7.2 - Rational Exponents - 7.2 Exercises: 64

Answer

$25$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the laws of exponents and the definition of rational exponents to simplify the given expression, $ \dfrac{125^{7/3}}{125^{5/3}} .$ $\bf{\text{Solution Details:}}$ 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} 125^{\frac{7}{3}-\frac{5}{3}} \\\\= 125^{\frac{2}{3}} .\end{array} Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \left( \sqrt[3]{125} \right)^{2} \\\\= \left( \sqrt[3]{(5)^3} \right)^{2} \\\\= \left( 5 \right)^{2} \\\\= 25 .\end{array}
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