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1. The value of $\sqrt{\frac{(\sqrt{12} - \sqrt{8}) (\sqrt{3} + \sqrt{2})}{5 + \sqrt{24}}}$ is = ?

A. $\sqrt{6} - \sqrt{2}$

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

C. $\sqrt{6} - 2$

D. $2 - \sqrt{6}$

Solution:

\begin{aligned} \sqrt{\frac{(\sqrt{12} - \sqrt{8}) (\sqrt{3} + \sqrt{2})}{5 + \sqrt{24}}} \\ = \sqrt{\frac{\sqrt{36} + \sqrt{24} - \sqrt{24} - \sqrt{16}}{5 + \sqrt{24}}} \\ = \sqrt{\frac{6 - 4}{5 + \sqrt{24}}} \\ = \sqrt{\frac{2}{5 + \sqrt{24}} \times \frac{5 - \sqrt{24}}{5 - \sqrt{24}}} \\ = \sqrt{\frac{2 (5 - \sqrt{24})}{25 - 24}} \\ = \sqrt{2 (5 - 2 \sqrt{6})} \\ = \sqrt{2 {{(\sqrt{3})}^{2} + {(\sqrt{2})}^{2} - 2 \sqrt{3} \times \sqrt{2}}} \\ = \sqrt{2 {(\sqrt{3} - \sqrt{2})}^{2}} \\ = \sqrt{2} (\sqrt{3} - \sqrt{2}) \\ = \sqrt{6} - 2 \end{aligned}

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