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1. The simplified form of $\frac{2}{\sqrt{7} + \sqrt{5}} +$ $\frac{7}{\sqrt{12} - \sqrt{5}} -$ $\frac{5}{\sqrt{12} - \sqrt{7}}$ is = ?

A. 5

B. 2

C. 1

D. 0

Solution:

\frac{2}{\sqrt{7} + \sqrt{5}} + \frac{7}{\sqrt{12} - \sqrt{5}} - \frac{5}{\sqrt{12} - \sqrt{7}}

= \frac{2}{\sqrt{7} + \sqrt{5}} \times

\frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} +

\frac{7}{\sqrt{12} - \sqrt{5}} \times

\frac{\sqrt{12} + \sqrt{5}}{\sqrt{12} + \sqrt{5}} -

(\frac{5}{\sqrt{12} - \sqrt{7}} \times \frac{\sqrt{12} + \sqrt{7}}{\sqrt{12} + \sqrt{7}})

= \frac{2 (\sqrt{7} - \sqrt{5})}{2} +

\frac{7 (\sqrt{12} + \sqrt{5})}{7} -

\frac{5 (\sqrt{12} + \sqrt{7})}{5}

\begin{aligned} = \sqrt{7} - \sqrt{5} + \sqrt{12} + \sqrt{5} - \sqrt{12} - \sqrt{7} \\ = 0 \end{aligned}

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