# How do you rationalize the denominator and simplify sqrt21/sqrt35?

Sep 6, 2017

$= \frac{\sqrt{15}}{5}$

#### Explanation:

multiply top and bottom by $\sqrt{35}$

$\frac{\sqrt{21}}{\sqrt{35}} \times \frac{\sqrt{35}}{\sqrt{35}}$

$= \frac{\sqrt{21} \sqrt{35}}{35}$

now combine the sqrt on the numerator and simplify as shown

$\frac{\sqrt{21 \times 35}}{35} = \frac{\sqrt{3 \times 7 \times 5 \times 7}}{35}$

$= \frac{\sqrt{{7}^{2} \times 15}}{35}$

$= \frac{\cancel{7} \sqrt{15}}{\cancel{35}} ^ 5$

$= \frac{\sqrt{15}}{5}$

Sep 6, 2017

Slightly different approach demonstrating that you can 'split' roots

$\frac{\sqrt{15}}{5}$

#### Explanation:

Given: $\frac{\sqrt{21}}{\sqrt{35}}$

Note that $5 \times 7 = 35 \mathmr{and} 3 \times 7 = 21$

Write as: $\frac{\sqrt{3} \times \cancel{\sqrt{7}}}{\sqrt{5} \times \cancel{\sqrt{7}}}$

Giving $\frac{\sqrt{3}}{\sqrt{5}}$

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

$\textcolor{g r e e n}{\frac{\sqrt{3}}{\sqrt{5}} \textcolor{red}{\times 1} \textcolor{w h i t e}{\text{ddd") ->color(white)("ddd}} \frac{\sqrt{3}}{\sqrt{5}} \textcolor{red}{\times \frac{\sqrt{5}}{\sqrt{5}}}} = \frac{\sqrt{15}}{5}$