# How do you simplify (-4+sqrt3)/(-1-2sqrt5)?

Jul 2, 2017

$\frac{- 4 + \sqrt{3}}{- 1 - 2 \sqrt{5}} = - \frac{4}{19} + \frac{8}{19} \sqrt{5} + \frac{\sqrt{3}}{19} - \frac{2}{19} \sqrt{15}$

#### Explanation:

When we have somethhing like $\sqrt{a} - \sqrt{b}$ in denominator, we simplify it by multiplying numerator and denominator by $\sqrt{a} + \sqrt{b}$. In case we have $\sqrt{a} + \sqrt{b}$ in denominator, multiply them by $\sqrt{a} - \sqrt{b}$.

In case we have $c - \sqrt{d}$ in denominator multiply by $c + \sqrt{d}$.

$\frac{- 4 + \sqrt{3}}{- 1 - 2 \sqrt{5}}$

= $\frac{- 4 + \sqrt{3}}{- 1 - 2 \sqrt{5}} \times \frac{- 1 + 2 \sqrt{5}}{- 1 + 2 \sqrt{5}}$

= $\frac{\left(- 4 + \sqrt{3}\right) \left(- 1 + 2 \sqrt{5}\right)}{{\left(- 1\right)}^{2} - {\left(2 \sqrt{5}\right)}^{2}}$

= $\frac{4 - 8 \sqrt{5} - \sqrt{3} + 2 \sqrt{15}}{1 - 20}$

= $\frac{4 - 8 \sqrt{5} - \sqrt{3} + 2 \sqrt{15}}{- 19}$

= $- \frac{4}{19} + \frac{8}{19} \sqrt{5} + \frac{\sqrt{3}}{19} - \frac{2}{19} \sqrt{15}$