Question

Find the value of $\frac{1}{\sqrt{5}-\sqrt{5-\sqrt{24}}}+\frac{1}{\sqrt{5}-\sqrt{5+\sqrt{24}}}$.

Answer 1

Now,

$\frac{1}{\sqrt{5}-\sqrt{5-\sqrt{24}}}+\frac{1}{\sqrt{5}-\sqrt{5+\sqrt{24}}}$

Let's rationalize the denominator of both the terms, we get

$=\frac{\sqrt{5}+\sqrt{5-\sqrt{24}}}{(\sqrt{5}{)}^2-(\sqrt{5-\sqrt{24}}{)}^2}+\frac{\sqrt{5}+\sqrt{5+\sqrt{24}}}{(\sqrt{5}{)}^2-(\sqrt{5+\sqrt{24}}{)}^2}$

$=\frac{1}{24}(\sqrt{5}+\sqrt{5-\sqrt{24}}-\sqrt{5}-\sqrt{5+\sqrt{24}})$

$=\frac{1}{24}(\sqrt{5-\sqrt{24}}-\sqrt{5+\sqrt{24}})$