# How do you rationalize the denominator and simplify 2/(1-sqrt5)?

Apr 28, 2018

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

#### Explanation:

To do this we must multiply both numerator and denominator by the denominators conjugate:

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

As this is just the same as multiplying by 1:

Expanding:

=> (2(1+sqrt(5) )) / (( 1-sqrt(5))(1+sqrt(5))

$\implies \frac{2 + 2 \sqrt{5}}{1 - \sqrt{5} + \sqrt{5} - \sqrt{5} \sqrt{5}}$

$\implies \frac{2 + 2 \sqrt{5}}{- 4}$

$\implies - \frac{1}{2} - \frac{1}{2} \sqrt{5} = - \frac{1 + \sqrt{5}}{2}$

Apr 28, 2018

$- \frac{1}{2} \left(1 + \sqrt{5}\right)$

#### Explanation:

$\text{to "color(blue)"rationalise the denominator}$

$\text{multiply the numerator/denominator by the "color(blue)"conjugate}$
$\text{of the denominator}$

$\text{the conjugate of "1-sqrt5" is } 1 \textcolor{red}{+} \sqrt{5}$

$\Rightarrow \frac{2 \left(1 + \sqrt{5}\right)}{\left(1 - \sqrt{5}\right) \left(1 + \sqrt{5}\right)}$

$\text{expand the denominator using FOIL}$

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