# Chapter 10 - Exponents and Radicals - 10.5 Expressions Containing Several Radical Terms - 10.5 Exercise Set: 83

$x^{}y^{}\sqrt[6]{xy^5 }$

#### Work Step by Step

Using $\sqrt[n]{x^m}=(\sqrt[n]{x})^m=x^{m/n}$, the given expression, $\sqrt[]{xy^3}\sqrt[3]{x^2y} ,$ is equivalent to \begin{array}{l}\require{cancel} (xy^3)^{\frac{1}{2}}\cdot (x^2y)^{\frac{1}{3}} .\end{array} Using the laws of exponents, the expression above simplifies to \begin{array}{l}\require{cancel} (xy^3)^{\frac{3}{6}}\cdot (x^2y)^{\frac{2}{6}} \\\\= \left[(xy^3)^{3}\cdot (x^2y)^{2} \right]^{\frac{1}{6}} \\\\= \sqrt[6]{(xy^3)^{3}\cdot (x^2y)^{2} } \\\\= \sqrt[6]{(x^3y^9)\cdot (x^4y^2) } \\\\= \sqrt[6]{x^{3+4}y^{9+2} } \\\\= \sqrt[6]{x^{7}y^{11} } \\\\= \sqrt[6]{x^{6}y^{6}\cdot xy^5 } \\\\= \sqrt[6]{(x^{}y^{})^6\cdot xy^5 } \\\\= x^{}y^{}\sqrt[6]{xy^5 } .\end{array} * Note that it is assumed that all variables represent positive numbers.

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