# How do you rationalize the denominator and simplify 3/(4+4sqrt5)?

Sep 26, 2015

Multiply both numerator and denominator by $\left(\sqrt{5} - 1\right)$ and rearrange to find:

$\frac{3}{4 + 4 \sqrt{5}} = \frac{3 \sqrt{5} - 3}{16}$

#### Explanation:

Using the identity ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$ we find:

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

=3/(4(1+sqrt(5))

=3/(4(sqrt(5)+1)

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

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

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

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

$= \frac{3 \left(\sqrt{5} - 1\right)}{16}$

$= \frac{3 \sqrt{5} - 3}{16}$