Question

The value of $\frac{1}{\sqrt{5}-\sqrt{5-\sqrt{24}}}+\frac{1}{\sqrt{5}-\sqrt{5+\sqrt{24}}}=$

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{3}$
• D. $\frac{-1}{\sqrt{3}}$

Answer 1

Answer: (D)

$\frac{-1}{\sqrt{3}}$

(D) $\frac{1}{\sqrt{5}-\sqrt{\sqrt{5}-\sqrt{24}}}+\frac{1}{\sqrt{5}-\sqrt{\sqrt{5}+\sqrt{24}}}$
$\frac{1}{\sqrt{5}-\sqrt{\sqrt{5}-2\sqrt{6}}}+\frac{1}{\sqrt{5}-\sqrt{\sqrt{5}+2\sqrt{6}}}$
$=\frac{1}{\sqrt{5}-(\sqrt{3}-\sqrt{2})}+\frac{1}{\sqrt{5}-(\sqrt{3}+\sqrt{2})}$
$=\frac{\sqrt{5}-\sqrt{3}-\sqrt{2}+\sqrt{5}-\sqrt{3}+\sqrt{2}}{(\sqrt{5}-\sqrt{3}{)}^2-(\sqrt{2}{)}^2}$

$=\frac{2(\sqrt{5}-\sqrt{3})}{8-2\sqrt{15}-2}=\frac{2(\sqrt{5}-\sqrt{3})}{2(3-\sqrt{15})}$
$=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{3}(\sqrt{3}-\sqrt{5})}=\frac{-1}{\sqrt{3}}$