## Algebra: A Combined Approach (4th Edition)

$\dfrac{xy}{\sqrt[3]{10xyz}}$
$\bf{\text{Solution Outline:}}$ To rationalize the numerator of the given expression, $\sqrt[3]{\dfrac{xy^2}{10z}} ,$ multiply by an expression equal to $1$ which will make the numerator a perfect power of the index. Then use the laws of radicals to simplify the resulting expression. $\bf{\text{Solution Details:}}$ Multiplying the given expression by an expression equal to $1$ which will make the numerator a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt[3]{\dfrac{xy^2}{10z}\cdot\dfrac{xy}{xy}} \\\\= \sqrt[3]{\dfrac{x^3y^3}{10xyz}} .\end{array} Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt[3]{x^3y^3}}{\sqrt[3]{10xyz}} \\\\= \dfrac{\sqrt[3]{(xy)^3}}{\sqrt[3]{10xyz}} \\\\= \dfrac{xy}{\sqrt[3]{10xyz}} .\end{array}