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1. $(\frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} + \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}})$ simplifies to = ?

A. $\sqrt{5} + \sqrt{6}$

B. $2 \sqrt{5} + \sqrt{6}$

C. $\sqrt{5} - \sqrt{6}$

D. $2 \sqrt{5} - 3 \sqrt{6}$

Solution:

\frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} + \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}}

\Rightarrow \frac{(1 + \sqrt{2}) (\sqrt{5} - \sqrt{3}) + (1 - \sqrt{2}) (\sqrt{5} + \sqrt{3})}{(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})}

\Rightarrow \frac{\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} + \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}}{5 - 3}

\begin{aligned} \Rightarrow \frac{2 \sqrt{5} - 2 \sqrt{6}}{2} \\ \Rightarrow \frac{2 (\sqrt{5} - \sqrt{6})}{2} \\ \Rightarrow \sqrt{5} - \sqrt{6} \end{aligned}

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