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1. Simplify : $\frac{1}{\sqrt{3} + \sqrt{4}} +$ $\frac{1}{\sqrt{4} + \sqrt{5}} +$ $\frac{1}{\sqrt{5} + \sqrt{6}} +$ $\frac{1}{\sqrt{6} + \sqrt{7}} +$ $\frac{1}{\sqrt{7} + \sqrt{8}} +$ $\frac{1}{\sqrt{8} + \sqrt{9}} = ?$

A. $\sqrt{3}$

B. $3 \sqrt{3}$

C. $3 - \sqrt{3}$

D. $5 - \sqrt{3}$

Solution:

\begin{aligned} \frac{1}{\sqrt{3} + \sqrt{4}} \\ = \frac{1}{\sqrt{4} + \sqrt{3}} \times \frac{\sqrt{4} - \sqrt{3}}{\sqrt{4} - \sqrt{3}} \\ = \frac{\sqrt{4} - \sqrt{3}}{1} \\ = \sqrt{4} - \sqrt{3} \\ Similarly \\ \frac{1}{\sqrt{4} + \sqrt{5}} = \sqrt{5} - \sqrt{4} \\ \Rightarrow \frac{1}{\sqrt{5} + \sqrt{6}} = \sqrt{6} - \sqrt{5} \\ \Rightarrow \frac{1}{\sqrt{6} + \sqrt{7}} = \sqrt{7} - \sqrt{6} \\ \Rightarrow \frac{1}{\sqrt{7} + \sqrt{8}} = \sqrt{8} - \sqrt{7} \\ \Rightarrow \frac{1}{\sqrt{8} + \sqrt{9}} = \sqrt{9} - \sqrt{8} \\ Now put values \\ \Rightarrow \sqrt{4} - \sqrt{3} + \sqrt{5} - \sqrt{4} + \sqrt{6} - \sqrt{5} + \sqrt{7} - \sqrt{6} + \sqrt{8} - \sqrt{7} + \sqrt{9} - \sqrt{8} \\ \Rightarrow \sqrt{9} - \sqrt{3} \\ \Rightarrow 3 - \sqrt{3} \end{aligned}

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