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

$\sqrt[6]{x^5}$
Using $a^{-x}=\dfrac{1}{a^x}$ or $\dfrac{1}{a^{-x}}=a^x,$ then \begin{array}{l}\require{cancel} \sqrt{x\sqrt[3]{x^2}} \\\\= \sqrt{x\cdot x^{2/3}} \\\\= (x\cdot x^{2/3})^{1/2} .\end{array} Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to \begin{array}{l}\require{cancel} (x\cdot x^{2/3})^{1/2} \\\\= \left( x^{1+\frac{2}{3}} \right)^{1/2} \\\\= \left( x^{\frac{3}{3}+\frac{2}{3}} \right)^{1/2} \\\\= \left( x^{\frac{5}{3}} \right)^{1/2} .\end{array} Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \left( x^{\frac{5}{3}} \right)^{1/2} \\\\= x^{\frac{5}{3}\cdot\frac{1}{2}} \\\\= x^{\frac{5}{6}} .\end{array} Using $x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x} \right)^m,$ then \begin{array}{l}\require{cancel} x^{\frac{5}{6}} \\\\= \sqrt[6]{x^5} .\end{array}