How do you simplify root3(4)/root5(8)?

Apr 8, 2017

$\frac{\sqrt[3]{4}}{\sqrt[5]{8}} = \sqrt[15]{2}$

Explanation:

Since we are dealing with positive radicands, we can freely combine the exponents like this:

$\frac{\sqrt[3]{4}}{\sqrt[5]{8}} = {\left({2}^{2}\right)}^{\frac{1}{3}} / {\left({2}^{3}\right)}^{\frac{1}{5}}$

$\textcolor{w h i t e}{\frac{\sqrt[3]{4}}{\sqrt[5]{8}}} = {2}^{\frac{2}{3}} / {2}^{\frac{3}{5}}$

$\textcolor{w h i t e}{\frac{\sqrt[3]{4}}{\sqrt[5]{8}}} = {2}^{\frac{2}{3} - \frac{3}{5}}$

$\textcolor{w h i t e}{\frac{\sqrt[3]{4}}{\sqrt[5]{8}}} = {2}^{\frac{10}{15} - \frac{9}{15}}$

$\textcolor{w h i t e}{\frac{\sqrt[3]{4}}{\sqrt[5]{8}}} = {2}^{\frac{1}{15}}$

$\textcolor{w h i t e}{\frac{\sqrt[3]{4}}{\sqrt[5]{8}}} = \sqrt[15]{2}$