# How do you simplify (4[sqrt5] )/( 7[sqrt2] - 7[sqrt5])?

Jun 8, 2016

$\frac{4 \sqrt{5}}{7 \sqrt{2} - 7 \sqrt{5}} = - \frac{20 + 4 \sqrt{10}}{21}$

#### Explanation:

$\frac{4 \sqrt{5}}{7 \sqrt{2} - 7 \sqrt{5}} = \frac{4 \sqrt{5}}{7 \left(\sqrt{2} - \sqrt{5}\right)}$

Now multiplying numerator and denominator by $\left(\sqrt{2} - \sqrt{5}\right)$, which is conjugate of denominator, we get

$\frac{4 \sqrt{5}}{7 \left(\sqrt{2} - \sqrt{5}\right)} \times \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} + \sqrt{5}}$

= $\frac{4 \sqrt{5} \left(\sqrt{2} + \sqrt{5}\right)}{7 \left(\sqrt{2} - \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}$

= $\frac{4 \sqrt{10} + 4 \cdot 5}{7 \left(2 - 5\right)}$

= $\frac{20 + 4 \sqrt{10}}{7 \cdot \left(- 3\right)}$

= $- \frac{20 + 4 \sqrt{10}}{21}$