# Chapter 10 - Exponents and Radicals - 10.5 Expressions Containing Several Radical Terms - 10.5 Exercise Set - Page 661: 99

$x\sqrt[6]{xy^{5}} - \sqrt[15]{x^{13}y^{14}}$

#### Work Step by Step

Using $a(b+c)=ab+ac$, or the Distributive Property, the given expression, $\sqrt[3]{x^2y} \left( \sqrt{xy}-\sqrt[5]{xy^3} \right) ,$ is equivalent to \begin{array}{l}\require{cancel} \sqrt[3]{x^2y} \left( \sqrt{xy} \right) - \sqrt[3]{x^2y} \left( \sqrt[5]{xy^3} \right) \end{array} Using the same indices for the radicals, the expression, $\sqrt[3]{x^2y} \left( \sqrt{xy} \right) - \sqrt[3]{x^2y} \left( \sqrt[5]{xy^3} \right)$, simplifies to \begin{array}{l}\require{cancel} \sqrt[3(2)]{x^{2(2)}y^{1(2)}} \left( \sqrt[2(3)]{x^{1(3)}y^{1(3)}} \right) - \sqrt[3(5)]{x^{2(5)}y^{1(5)}} \left( \sqrt[5(3)]{x^{1(3)}y^{3(3)}} \right) \\\\= \sqrt[6]{x^{4}y^{2}} \left( \sqrt[6]{x^{3}y^{3}} \right) - \sqrt[15]{x^{10}y^{5}} \left( \sqrt[15]{x^{3}y^{9}} \right) \\\\= \sqrt[6]{x^{4}y^{2}(x^{3}y^{3})} - \sqrt[15]{x^{10}y^{5}(x^{3}y^{9})} \\\\= \sqrt[6]{x^{4+3}y^{2+3}} - \sqrt[15]{x^{10+3}y^{5+9}} \\\\= \sqrt[6]{x^{7}y^{5}} - \sqrt[15]{x^{13}y^{14}} \\\\= \sqrt[6]{x^{6}\cdot xy^{5}} - \sqrt[15]{x^{13}y^{14}} \\\\= x\sqrt[6]{xy^{5}} - \sqrt[15]{x^{13}y^{14}} \end{array} * Note that it is assumed that all variables represent positive numbers.

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