Question

Find value $\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$ is

• A. 1
• B. 0
• C. $\sqrt{2}$
• D. None of these

Answer 1

The correct option is D None of these

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

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

$\Rightarrow \frac{2(\sqrt{2}-\sqrt{5})}{2+5-2\sqrt{10}-3}$

$=\frac{2(\sqrt{2}-\sqrt{5})}{4-2\sqrt{10}}$

$\Rightarrow \frac{\sqrt{2}-\sqrt{5}}{2-\sqrt{10}}=\frac{\sqrt{2}-\sqrt{5}}{\sqrt{2}(\sqrt{2}-\sqrt{5})}=\frac{1}{\sqrt{2}}$.