# How do you rationalize the denominator and simplify sqrt(10/7)?

Jul 9, 2017

$\frac{\sqrt{70}}{7}$

#### Explanation:

$\sqrt{\frac{10}{7}} = \frac{\sqrt{10}}{\sqrt{7}}$

To rationalise this denominatir, we multiply the top and bottom by $\sqrt{7}$, so we get $\frac{\sqrt{10}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$, $\sqrt{7} \cdot \sqrt{7}$ is just $7$, and $\sqrt{10} \cdot \sqrt{7} = \sqrt{10 \cdot 7} = \sqrt{70}$, as $70$ doesn't have any factors which are perfect squares, it can only stay as $\sqrt{70}$, giving us $\frac{\sqrt{70}}{7}$

Jul 9, 2017

See the solution process below:

#### Explanation:

First, we need to rewrite the expression using this rule for dividing radicals:

$\sqrt{\frac{\textcolor{red}{a}}{\textcolor{b l u e}{b}}} = \frac{\sqrt{\textcolor{red}{a}}}{\sqrt{\textcolor{b l u e}{b}}}$

$\sqrt{\frac{\textcolor{red}{10}}{\textcolor{b l u e}{7}}} = \frac{\sqrt{\textcolor{red}{10}}}{\sqrt{\textcolor{b l u e}{7}}}$

Next, we can rationalize the denominator and remove the radical by multiplying the fraction by the appropriate form of $1$:

$\frac{\sqrt{7}}{\sqrt{7}} \times \frac{\sqrt{10}}{\sqrt{7}} = \frac{\sqrt{7} \times \sqrt{10}}{\sqrt{7} \times \sqrt{7}} = \frac{\sqrt{7} \times \sqrt{10}}{7}$

We can now use this rule of exponents to simplify the numerator:

$\sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}}$

$\frac{\sqrt{\textcolor{red}{7}} \cdot \sqrt{\textcolor{b l u e}{10}}}{7} = \frac{\sqrt{\textcolor{red}{7} \cdot \textcolor{b l u e}{10}}}{7} = \frac{\sqrt{70}}{7}$