# What is (sqrt7- 5)/(sqrt2 + 5 )?

Jun 12, 2015

The answer is $- \frac{\sqrt{14} - 5 \sqrt{7} - 5 \sqrt{2} + 25}{23}$.

#### Explanation:

$\frac{\sqrt{7} - 5}{\sqrt{2} + 5}$

Rationalize the denominator by multiplying the numerator and denominator by $\left(\sqrt{2} - 5\right)$.

$\frac{\sqrt{7} - 5}{\sqrt{2} + 5} \cdot \frac{\sqrt{2} - 5}{\sqrt{2} - 5}$ =

$\frac{\left(\sqrt{7} - 5\right) \left(\sqrt{2} - 5\right)}{\left(\sqrt{2} + 5\right) \left(\sqrt{2} - 5\right)}$ =

The denominator is in the form of the difference of squares: ${a}^{2} - {b}^{2}$

$\frac{\left(\sqrt{7} - 5\right) \left(\sqrt{2} - 5\right)}{{\left(\sqrt{2}\right)}^{2} - {5}^{2}}$ =

$\frac{\left(\sqrt{7} - 5\right) \left(\sqrt{2} - 5\right)}{2 - 25}$ =

$\frac{\left(\sqrt{7} - 5\right) \left(\sqrt{2} - 5\right)}{- 23}$

FOIL the numerator.

$\frac{\sqrt{14} - 5 \sqrt{7} - 5 \sqrt{2} + 25}{- 23}$ =

$- \frac{\sqrt{14} - 5 \sqrt{7} - 5 \sqrt{2} + 25}{23}$