# How do you simplify root3(3/4)?

Apr 23, 2017

$\sqrt[3]{\frac{3}{4}} = \frac{\sqrt[3]{6}}{2}$

#### Explanation:

For any non-zero values of $a , b$ we have:

$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$

$\sqrt[3]{{a}^{3}} = a$

So we find:

$\sqrt[3]{\frac{3}{4}} = \sqrt[3]{\frac{3 \cdot 2}{4 \cdot 2}} = \sqrt[3]{\frac{6}{2} ^ 3} = \frac{\sqrt[3]{6}}{\sqrt[3]{{2}^{3}}} = \frac{\sqrt[3]{6}}{2}$

Notice how making the denominator into a perfect cube before splitting the radical allows us to avoid having to rationalise the denominator afterwards.

Apr 23, 2017

color(blue)(root3(6)/2

#### Explanation:

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

$\therefore = \frac{\sqrt[3]{3}}{\sqrt[3]{4}} \times \frac{\sqrt[3]{4}}{\sqrt[3]{4}} \times \frac{\sqrt[3]{4}}{\sqrt[3]{4}}$

:.=color(blue)(root3(4)*root3(4)*root3(4)=4

$\therefore = \frac{\sqrt[3]{48}}{4}$

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

$\therefore = \frac{{\cancel{2}}^{\textcolor{b l u e}{1}} \sqrt[3]{6}}{\cancel{4}} ^ \textcolor{b l u e}{2}$

:.=color(blue)(root3(6)/2