# How do you simplify \frac { 9} { \sqrt { 21} }?

Apr 3, 2017

$\frac{27}{7}$

#### Explanation:

Firstly, we have to get rid of the radical. To do that, we can $^ 2$(square) the entire fraction and then solve from there.

${\left(\frac{9}{\sqrt{21}}\right)}^{2} = \frac{9 \cdot 9}{\sqrt{21}} ^ 2 = \frac{81}{21}$ ~(3 can be taken out)~ $= \frac{27}{7}$

Apr 3, 2017

$\frac{3 \sqrt{21}}{7}$

#### Explanation:

To eliminate the $\sqrt{21}$ from the denominator of the fraction we use a method called $\textcolor{b l u e}{\text{rationalising}}$

This ensures that we have a $\textcolor{b l u e}{\text{rational}}$ denominator as opposed to a $\textcolor{b l u e}{\text{surd}} .$

Consider $\sqrt{100} \times \sqrt{100}$

$\sqrt{100} = 10$

$\Rightarrow \sqrt{100} \times \sqrt{100}$

$= 10 \times 10$

$= 100 \leftarrow \textcolor{red}{\text{ the value inside the root}}$

$\text{in general } \sqrt{a} \times \sqrt{a} = a$

$\text{Since " 9/sqrt21" is a fraction}$ we multiply the numerator/denominator by $\sqrt{21}$

$\Rightarrow \frac{9}{\sqrt{21}} \times \frac{\sqrt{21}}{\sqrt{21}}$

$= \frac{9 \times \sqrt{21}}{21} \leftarrow \textcolor{red}{\text{ using above result}}$

$= \frac{{\cancel{9}}^{3} \times \sqrt{21}}{\cancel{21}} ^ 7 \leftarrow \textcolor{red}{\text{ cancelling by 3}}$

$= \frac{3 \sqrt{21}}{7}$

Apr 5, 2017

color(red)(=(3sqrt21)/7

#### Explanation:

$\frac{9}{\sqrt{21}}$

$\therefore = {3}^{2} / \left(\sqrt{3} \times \sqrt{7}\right)$

$\therefore = {3}^{2} / \left({3}^{\frac{1}{2}} \sqrt{7}\right)$

$\therefore = {3}^{2 - \frac{1}{2}} / \sqrt{7}$

$\therefore = \frac{{3}^{1 \frac{1}{2}}}{\sqrt{7}}$

$\therefore = {3}^{\frac{3}{2}} / \sqrt{7}$

$\therefore = \frac{\sqrt{{3}^{3}}}{\sqrt{7}}$

$\therefore = \frac{\sqrt{{3}^{3}}}{\sqrt{7}}$

Rationalize denominator by multiplying by $\frac{\sqrt{7}}{\sqrt{7}}$

$\therefore = \frac{\sqrt{{3}^{3}}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}$

$\therefore = \sqrt{3} \cdot \sqrt{3} = 3 , \sqrt{7} \cdot \sqrt{7} = 7$

$\therefore = \frac{\sqrt{3 \cdot 3 \cdot 3 \cdot 7}}{7}$

$\therefore = \frac{3 \sqrt{3 \cdot 7}}{7}$

:.color(red)(=(3sqrt21)/7