## Intermediate Algebra (12th Edition)

$\sqrt[8]{9^5q^5}-\sqrt[3]{4x^2}$
$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents to convert the given expression, $(9q)^{5/8}-(2x)^{2/3} ,$ to radical form. Then use the laws of exponents to simplify the resulting expression. $\bf{\text{Solution Details:}}$ Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \sqrt[8]{(9q)^5}-\sqrt[3]{(2x)^2} .\end{array} Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \sqrt[8]{9^5q^5}-\sqrt[3]{2^2x^2} \\\\= \sqrt[8]{9^5q^5}-\sqrt[3]{4x^2} .\end{array}