# How do you rationalize the denominator and simplify 1/{1+sqrt(3)-sqrt(5)}?

Apr 2, 2018

Attempt to make the three term denominator two terms..

#### Explanation:

Multiply the fraction by $\frac{1 + \sqrt{3} + \sqrt{5}}{1 + \sqrt{3} + \sqrt{5}}$ . This is the same as multiplying by 1. This is two eliminate two surds.
The denominator you should get after multiplying the two fractions is
$\left(1 + \sqrt{3} + \sqrt{5}\right) \cdot \left(1 + \sqrt{3} - \sqrt{5}\right)$ . We can write this as ((1+sqrt3)-sqrt5)(1+sqrt3)+sqrt5).
This can then be made is DOPS and written as : ${\left(1 + \sqrt{3}\right)}^{2} - {\left(\sqrt{5}\right)}^{2}$.
Simplify that to get: $1 + 3 + 2 \sqrt{3} - 5$
I'll leave it here :).

Apr 2, 2018

$\frac{1}{1 + \sqrt{3} - \sqrt{5}} = \frac{7 + 3 \sqrt{3} + \sqrt{5} + 2 \sqrt{15}}{11}$

#### Explanation:

This involves two stages of rationalisation to get rid of terms in $\sqrt{3}$ and $\sqrt{5}$. Both steps use the difference of squares identity:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

So:

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

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{1 + \sqrt{3} + \sqrt{5}}{{\left(1 + \sqrt{3}\right)}^{2} - {\left(\sqrt{5}\right)}^{2}}$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{1 + \sqrt{3} + \sqrt{5}}{1 + 2 \sqrt{3} + 3 - 5}$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{1 + \sqrt{3} + \sqrt{5}}{2 \sqrt{3} - 1}$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{\left(1 + \sqrt{3} + \sqrt{5}\right) \left(2 \sqrt{3} + 1\right)}{\left(2 \sqrt{3} - 1\right) \left(2 \sqrt{3} + 1\right)}$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{\left(1 + \sqrt{3} + \sqrt{5}\right) \left(2 \sqrt{3} + 1\right)}{{\left(2 \sqrt{3}\right)}^{2} - {1}^{2}}$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{\left(1 + \sqrt{3} + \sqrt{5}\right) \left(2 \sqrt{3} + 1\right)}{12 - 1}$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{1}{11} \left(1 + \sqrt{3} + \sqrt{5}\right) \left(2 \sqrt{3} + 1\right)$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{1}{11} \left(2 \sqrt{3} \left(1 + \sqrt{3} + \sqrt{5}\right) + 1 \left(1 + \sqrt{3} + \sqrt{5}\right)\right)$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{1}{11} \left(2 \sqrt{3} + 6 + 2 \sqrt{15} + 1 + \sqrt{3} + \sqrt{5}\right)$

$\textcolor{w h i t e}{\frac{1}{1 + \sqrt{3} - \sqrt{5}}} = \frac{7 + 3 \sqrt{3} + \sqrt{5} + 2 \sqrt{15}}{11}$