Question : $\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}=?$
Option 1: 5
Option 2: 4
Option 3: 3
Option 4: 2
Correct Answer: 5
Solution :
$\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}$
Rationalise the denominators,
$=(\frac{\sqrt{8}+3}{1})-(\frac{\sqrt{7}+\sqrt{8}}{1})+(\frac{\sqrt{6}+\sqrt{7}}{1})-(\frac{\sqrt{5}+\sqrt{6}}{1})+(\frac{2+\sqrt{5}}{1}$)
$=\sqrt{8}+3-\sqrt{8}-\sqrt{7}+\sqrt{7}+\sqrt{6}-\sqrt{6}-\sqrt{5}+\sqrt{5}+2$
$=3+2$
$= 5$
Hence, the correct answer is 5.
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