# How do you simplify 4(sqrt2-sqrt7)?

Jul 3, 2017

$4 \left(\sqrt{2} - \sqrt{7}\right) = 4 \sqrt{2} - 4 \sqrt{7}$

#### Explanation:

The two square roots $\sqrt{2}$ and $\sqrt{7}$ are essentially unrelated to one another. So about all we can do to simplify the given expression is multiply it out to get:

$4 \left(\sqrt{2} - \sqrt{7}\right) = 4 \sqrt{2} - 4 \sqrt{7}$

$\textcolor{w h i t e}{}$
Bonus

If you encountered $4 \left(\sqrt{2} - \sqrt{7}\right)$ as the denominator of a rational expression, then you could rationalise the denominator by multiplying by $\sqrt{2} + \sqrt{7}$. For example:

$\frac{1}{4 \left(\sqrt{2} - \sqrt{7}\right)} = \frac{\sqrt{2} + \sqrt{7}}{4 \left(\sqrt{2} - \sqrt{7}\right) \left(\sqrt{2} + \sqrt{7}\right)}$

$\textcolor{w h i t e}{\frac{1}{4 \left(\sqrt{2} - \sqrt{7}\right)}} = \frac{\sqrt{2} + \sqrt{7}}{4 \left({\left(\sqrt{2}\right)}^{2} - {\left(\sqrt{7}\right)}^{2}\right)}$

$\textcolor{w h i t e}{\frac{1}{4 \left(\sqrt{2} - \sqrt{7}\right)}} = \frac{\sqrt{2} + \sqrt{7}}{4 \left(2 - 7\right)}$

$\textcolor{w h i t e}{\frac{1}{4 \left(\sqrt{2} - \sqrt{7}\right)}} = \frac{\sqrt{2} + \sqrt{7}}{- 20}$

$\textcolor{w h i t e}{\frac{1}{4 \left(\sqrt{2} - \sqrt{7}\right)}} = - \frac{\sqrt{2}}{20} - \frac{\sqrt{7}}{20}$