Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 10 - Exponents and Radicals - 10.5 Expressions Containing Several Radical Terms - 10.5 Exercise Set: 37

Answer

$3-4\sqrt[3]{63}$

Work Step by Step

Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the given expression is equivalent to \begin{array}{l}\require{cancel} \sqrt[3]{3}(\sqrt[3]{9}-4\sqrt[3]{21}) \\\\= \sqrt[3]{3}(\sqrt[3]{9})+\sqrt[3]{3}(-4\sqrt[3]{21}) .\end{array} Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \sqrt[3]{3}(\sqrt[3]{9})+\sqrt[3]{3}(-4\sqrt[3]{21}) \\\\= \sqrt[3]{3(9)}-4\sqrt[3]{3(21)} \\\\= \sqrt[3]{27}-4\sqrt[3]{63} .\end{array} Extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt[3]{27}-4\sqrt[3]{63} \\\\= \sqrt[3]{(3)^3}-4\sqrt[3]{63} \\\\= 3-4\sqrt[3]{63} .\end{array}
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