## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\sqrt[6]{7}$
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the given expression is equivalent to \begin{array}{l}\require{cancel} \dfrac{7^{-1/3}}{7^{-1/2}} \\\\= 7^{-\frac{1}{3}-\left( -\frac{1}{2} \right)} \\\\= 7^{-\frac{1}{3}+\frac{1}{2}} \\\\= 7^{-\frac{2}{6}+\frac{3}{6}} \\\\= 7^{\frac{1}{6}} .\end{array} 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 given expression, $8^{2/3} ,$ is equivalent to \begin{array}{l}\require{cancel} 7^{\frac{1}{6}} \\\\= \sqrt[6]{7^1} \\\\= \sqrt[6]{7} .\end{array}