# Chapter 10 - Exponents and Radicals - 10.5 Expressions Containing Several Radical Terms - 10.5 Exercise Set: 38

$-6$

#### 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]{2}(\sqrt[3]{4}-2\sqrt[3]{32}) \\\\= \sqrt[3]{2}(\sqrt[3]{4})+\sqrt[3]{2}(-2\sqrt[3]{32}) .\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]{2}(\sqrt[3]{4})+\sqrt[3]{2}(-2\sqrt[3]{32}) \\\\= \sqrt[3]{2(4)}-2\sqrt[3]{2(32)} \\\\= \sqrt[3]{8}-2\sqrt[3]{64} .\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]{8}-2\sqrt[3]{64} \\\\= \sqrt[3]{(2)^3}-2\sqrt[3]{(4)^3} \\\\= 2-2(4) \\\\= 2-8 \\\\= -6 .\end{array}

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