# How do you simplify (2+5sqrt3)/(-4+4sqrt2)?

Nov 7, 2016

$\frac{2 + 2 \sqrt{2} + 5 \sqrt{3} + 5 \sqrt{6}}{4}$

#### Explanation:

In surd form we must not have surds (roots) in the denominator.
We use the fact that $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$

The denominator factorises to $4 \left(- 1 + \sqrt{2}\right)$
So we multiply both numerator and denominator by the same thing $\left(- 1 - \sqrt{2}\right)$

$\frac{2 + 5 \sqrt{3}}{4 \left(- 1 + \sqrt{2}\right)}$ =$\frac{2 + 5 \sqrt{3}}{4 \left(- 1 + \sqrt{2}\right)} \cdot \left(\frac{- 1 - \sqrt{2}}{- 1 - \sqrt{2}}\right)$
Numerator$\left(2 + 5 \sqrt{3}\right) \left(- 1 - \sqrt{2}\right) = - 2 - 2 \sqrt{2} - 5 \sqrt{3} - 5 \sqrt{6}$
or -(2+2sqrt2+5sqrt3+5sqrt6)

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