Question

Simplify:

$\frac{1}{\sqrt{12-2\sqrt{35}}}+\frac{1}{\sqrt{8-2\sqrt{15}}}-\frac{2}{\sqrt{10-2\sqrt{21}}}$.

Answer 1

$\frac{1}{\sqrt{12-2\sqrt{35}}}+\frac{1}{\sqrt{8-2\sqrt{15}}}-\frac{2}{\sqrt{10-2\sqrt{21}}}$

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

$\Rightarrow \frac{1}{\sqrt{(\sqrt{7}-\sqrt{5}{)}^2}}+\frac{1}{\sqrt{(\sqrt{5}-\sqrt{3}{)}^2}}-\frac{2}{\sqrt{(\sqrt{7}-\sqrt{3}{)}^2}}$

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

$\Rightarrow \frac{1(\sqrt{7}+\sqrt{5})}{(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5})}+\frac{1(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}-\frac{2(\sqrt{7}+\sqrt{3})}{(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{3})}$

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

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

The answer is $\sqrt{5}$