A) \[\sqrt{6}-\sqrt{2}\]
B) \[\sqrt{6}+\sqrt{2}\]
C) \[\sqrt{6}-2\]
D) \[2-\sqrt{6}\]
Correct Answer: C
Solution :
\[\sqrt{\frac{(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})}{5+\sqrt{24}}}\] \[=\sqrt{\frac{\sqrt{36}-\sqrt{24}+\sqrt{24}-\sqrt{16}}{5+\sqrt{24}}}\] \[=\sqrt{\frac{6-4}{5+\sqrt{24}}}=\sqrt{\frac{2}{5+\sqrt{24}}}\] \[=\sqrt{\frac{2}{5+\sqrt{6\times 4}}}=\sqrt{\frac{2}{5+2\sqrt{6}}}\] \[=\sqrt{\frac{2}{5+2\sqrt{6}}\times \frac{5-2\sqrt{6}}{5-2\sqrt{6}}}\] \[=\sqrt{\frac{2\,\,(5-2\sqrt{6})}{25-24}}=\sqrt{2\,\,(5-2\sqrt{6})}\] \[=\sqrt{2\,\,[{{(\sqrt{3})}^{2}}+{{(\sqrt{2})}^{2}}-2\sqrt{3}\sqrt{2}]}\] \[=\sqrt{2\,\,{{(\sqrt{3}-\sqrt{2})}^{2}}}=\sqrt{2}(\sqrt{3}-\sqrt{2})=\sqrt{6}-2\]
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