Question

Find the value of : $\sqrt{\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-7}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{2}-2}}$

Answer 1

$\frac{1}{3-\sqrt{8}}=\frac{3+\sqrt{8}}{(3-\sqrt{8})(3+\sqrt{8})}=\frac{3+\sqrt{8}}{9-8}=3+\sqrt{8}=\frac{1}{\sqrt{8}-\sqrt{7}}=\frac{\sqrt{8}+\sqrt{7}}{8-7}=\sqrt{8}+\sqrt{7}=\frac{1}{\sqrt{7}-sqrt6}=\sqrt{7}+\sqrt{6}=\frac{1}{\sqrt{6}-\sqrt{5}}=\sqrt{6}+\sqrt{5}=\frac{1}{\sqrt{2}-2}=\frac{\sqrt{2}+2}{2-4}=-\frac{(2+\sqrt{2})}{2}\therefore \sqrt{(}3+\sqrt{8}-\sqrt{8}-\sqrt{7}+\sqrt{7}+\sqrt{6}-\sqrt{6}-\sqrt{5}-\sqrt{5}-1-\frac{1}{\sqrt{5}})=\sqrt{(}2-\sqrt{5}-\frac{1}{\sqrt{2}})$