# How do you simplify sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3)))-sqrt((sqrt(3)-1)^2) ?

Oct 22, 2017

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

#### Explanation:

Given:

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

First note that:

${\left(\sqrt{3 - \sqrt{5 + 2 \sqrt{3}}} + \sqrt{3 + \sqrt{5 + 2 \sqrt{3}}}\right)}^{2}$

$= \left(3 - \sqrt{5 + 2 \sqrt{3}}\right) + 2 \sqrt{9 - \left(5 + 2 \sqrt{3}\right)} + \left(3 + \sqrt{5 + 2 \sqrt{3}}\right)$

$= 6 + 2 \sqrt{4 - 2 \sqrt{3}}$

$= 6 + 2 \sqrt{3 - 2 \sqrt{3} + 1}$

$= 6 + 2 \sqrt{{\left(\sqrt{3} - 1\right)}^{2}}$

$= 6 + 2 \left(\sqrt{3} - 1\right)$

$= 4 + 2 \sqrt{3}$

$= 3 + 2 \sqrt{3} + 1$

$= {\left(\sqrt{3} + 1\right)}^{2}$

So:

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

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

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

$= 2$