# How do you rationalize the denominator and simplify 1/root3(x^2)?

Oct 17, 2017

$\frac{1}{\sqrt{{x}^{2}}} = \textcolor{red}{{x}^{- \frac{4}{3}}}$

#### Explanation:

$\frac{1}{\sqrt{{x}^{2}}}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{1}{\sqrt{{x}^{2}}} \times \frac{\sqrt{{x}^{2}}}{\sqrt{{x}^{2}}} \times \frac{\sqrt{{x}^{2}}}{\sqrt{{x}^{2}}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{{\left(\sqrt{{x}^{2}}\right)}^{2}}{{x}^{2}}$

color(white)("XXX")=(x^(1/3))^2xx`1/(x^2)

$\textcolor{w h i t e}{\text{XXX}} = {x}^{\frac{2}{3}} \times {x}^{- 2}$

$\textcolor{w h i t e}{\text{XXX}} = {x}^{\left(\frac{2}{3} - 2\right)}$

$\textcolor{w h i t e}{\text{XXX}} = {x}^{\left(- \frac{4}{3}\right)}$

Oct 17, 2017

See a solution process below:

#### Explanation:

To rationalize the denominator we must multiply the fraction by the appropriate value of $1$:

$\frac{\sqrt{x}}{\sqrt{x}} \times \frac{1}{\sqrt{{x}^{2}}} \implies$

$\frac{\sqrt{x} \times 1}{\sqrt{x} \times \sqrt{{x}^{2}}} \implies$

$\frac{\sqrt{x}}{\sqrt{x \times {x}^{2}}} \implies$

$\frac{\sqrt{x}}{\sqrt{{x}^{3}}} \implies$

$\frac{\sqrt{x}}{x}$