How do you simplify root3(x) / root 3(2)?

2 Answers
Apr 15, 2017

$\sqrt[3]{\frac{x}{2}}$

Explanation:

$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Apr 15, 2017

$\frac{\sqrt[3]{4 x}}{2}$

Explanation:

Usually, you do not want a radical in the denominator. In other words, you usually want to rationalize the denominator.

Here, the fraction can be expressed as ${x}^{\frac{1}{3}} / {2}^{\frac{1}{3}}$. If you multiply this by ${2}^{\frac{2}{3}} / {2}^{\frac{2}{3}}$ (which is equal to $1$), the denominator is rationalized. To see how, remember that ${a}^{b} \cdot {a}^{c} = {a}^{b + c}$. Then, ${x}^{\frac{3}{2}} / {2}^{\frac{3}{2}} \cdot {2}^{\frac{1}{2}} / {2}^{\frac{1}{2}} = \frac{{x}^{\frac{1}{3}} \cdot {2}^{\frac{2}{3}}}{{2}^{\frac{1}{3}} \cdot {2}^{\frac{2}{3}}} = \frac{{x}^{\frac{1}{3}} \cdot {2}^{\frac{2}{3}}}{{2}^{\frac{1}{3} + \frac{2}{3}}} = \frac{{x}^{\frac{1}{3}} \cdot {2}^{\frac{2}{3}}}{4} = \frac{{x}^{\frac{1}{3}} \cdot {2}^{\frac{2}{3}}}{2} = \frac{\sqrt[3]{x} \cdot \sqrt[3]{4}}{2} = \frac{\sqrt[3]{4 x}}{2}$.