Question

Prove that $\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}=5$

Answer 1

$\frac{1}{\sqrt{a+1}+\sqrt{a}}$

$=\frac{\sqrt{a+1}+\sqrt{a}}{a+1-a}$

$=\sqrt{a+1}-\sqrt{a}$

$\Rightarrow \frac{1}{3-\sqrt{a}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}+\frac{1}{\sqrt{5}-2}-\frac{1}{\sqrt{6}-\sqrt{5}}$

$=3+\sqrt{8}-\sqrt{8}-\sqrt{7}+\sqrt{7}+\sqrt{6}+\sqrt{5}+2-\sqrt{6}-\sqrt{5}$

$=3+2$

$=5$