Question

Show 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

We have,

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

L.H.S.

$\frac{1}{(3-\sqrt{8})}-\frac{1}{(\sqrt{8}-\sqrt{7})}+\frac{1}{(\sqrt{7}-\sqrt{6})}-\frac{1}{(\sqrt{6}-\sqrt{5})}+\frac{1}{(\sqrt{5}-2)}$

By rationalize every part and we get

$\frac{1}{(3-\sqrt{8})}\times \frac{(3+\sqrt{8})}{(3+\sqrt{8})}-\frac{1}{(\sqrt{8}-\sqrt{7})}\times \frac{(\sqrt{8}+\sqrt{7})}{(\sqrt{8}+\sqrt{7})}+\frac{1}{(\sqrt{7}-\sqrt{6})}\times \frac{(\sqrt{7}+\sqrt{6})}{(\sqrt{7}+\sqrt{6})}-\frac{1}{(\sqrt{6}-\sqrt{5})}\times \frac{(\sqrt{6}+\sqrt{5})}{(\sqrt{6}+\sqrt{5})}+\frac{1}{(\sqrt{5}-2)}\times \frac{(\sqrt{5}+2)}{(\sqrt{5}+2)}$

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

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

$=3+2=5$

R.H.S

Hence proved.