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

Oct 16, 2015

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

#### Explanation:

You're going to have to do a little work here to simplify this expression.

The way to go is by rationalizing the denominator. The only problem is the fact that your denominator is a trinomial, and conjugates are only formed for binomials.

More specifically, you get the conjugate of a binomial by changing the sign of its second term.

a + b -> underbrace(a color(red)(-) b)_(color(blue)("conjugate"))" " or $\text{ "a - b -> overbrace(a color(red)(+) b)^(color(blue)("conjugate"))" }$

This means that you're going to have to group the denominator as a binomial, for which you can write

${\overbrace{1}}^{\textcolor{red}{a}} + {\overbrace{\left(\sqrt{3} - \sqrt{5}\right)}}^{\textcolor{red}{b}} \to {\underbrace{1 \textcolor{red}{-} \left(\sqrt{3} - \sqrt{5}\right)}}_{\textcolor{b l u e}{\text{conjugate}}}$

So, multiply your expression by $1 = \frac{1 - \left(\sqrt{3} - \sqrt{5}\right)}{1 - \left(\sqrt{3} - \sqrt{5}\right)}$ to get

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

(1 - sqrt(3) + sqrt(5))/([1 + (sqrt(3) - sqrt(5))][1 - (sqrt(3) - sqrt(5))]

The denominator can be rewritten as

$\left[1 + \left(\sqrt{3} - \sqrt{5}\right)\right] \left[1 - \left(\sqrt{3} - \sqrt{5}\right)\right] = {1}^{2} - {\left(\sqrt{3} - \sqrt{5}\right)}^{2}$

This, in turn, will be equal to

$1 - \left({\left(\sqrt{3}\right)}^{2} - 2 \sqrt{3 \cdot 5} + {\left(\sqrt{5}\right)}^{2}\right) = 1 - 3 + 2 \sqrt{15} - 5$

$= 2 \sqrt{15} - 7$

The expression becomes

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

Now do the same thing with the new denominator, i.e. find its conjugate

$2 \sqrt{15} - 7 \to 2 \sqrt{15} \textcolor{red}{+} 7$

and multiply the expression by $1 = \frac{2 \sqrt{15} + 7}{2 \sqrt{15} + 7}$ to get

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

$\frac{\left(1 - \sqrt{3} + \sqrt{5}\right) \left(2 \sqrt{15} + 7\right)}{\left(2 \sqrt{15} - 7\right) \left(2 \sqrt{15} + 7\right)}$

The denominator will be equal to

$\left(2 \sqrt{15} - 7\right) \left(2 \sqrt{15} + 7\right) = {\left(2 \sqrt{15}\right)}^{2} - {7}^{2}$

$= 4 \cdot 15 - 49 = 11$

The numerator will be

$\left(1 - \sqrt{3} + \sqrt{5}\right) \left(2 \sqrt{15} + 7\right)$

$2 \sqrt{15} - 2 \sqrt{45} + 2 \sqrt{75} + 7 - 7 \sqrt{3} + 7 \sqrt{5}$

$2 \sqrt{15} - 6 \sqrt{5} + 10 \sqrt{3} + 7 - 7 \sqrt{3} + 7 \sqrt{5}$

$7 + 3 \sqrt{3} + \sqrt{5} + 2 \sqrt{15}$

The simplified expression will thus be

$\frac{1}{1 + \sqrt{3} - \sqrt{5}} = \textcolor{g r e e n}{\frac{7 + 3 \sqrt{3} + \sqrt{5} + 2 \sqrt{15}}{11}}$