How do you rationalize the denominator and simplify 5/(root3(4))?

Apr 19, 2017

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

Explanation:

Multiply the fraction by $\frac{{4}^{\frac{2}{3}}}{{4}^{\frac{2}{3}}}$:

$\frac{5}{\sqrt[3]{4}} \textcolor{b l u e}{\times \frac{\sqrt[3]{4}}{\sqrt[3]{4}} \times \frac{\sqrt[3]{4}}{\sqrt[3]{4}}}$

$= \frac{5 \cdot {4}^{\frac{2}{3}}}{4}$

$= \frac{5 \sqrt[3]{16}}{4}$

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

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

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

Apr 19, 2017

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

Explanation:

Given:

$\frac{5}{\sqrt[3]{4}}$

First note that:

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

So to make the denominator rational it will be sufficient to multiply it by $\sqrt[3]{2}$...

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

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

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

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