# How do you divide (sqrt5+1)/(sqrt5-1)?

Jun 6, 2015

Multiply both numerator (top) and denominator (bottom) by the conjugate $\left(\sqrt{5} + 1\right)$ of the denominator:

$\frac{\sqrt{5} + 1}{\sqrt{5} - 1} = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \cdot \frac{\sqrt{5} + 1}{\sqrt{5} + 1}$

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

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

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

$= \frac{6 + 2 \sqrt{2}}{4}$

$= \frac{2 \left(3 + \sqrt{5}\right)}{2 \cdot 2}$

$= \frac{3 + \sqrt{5}}{2}$