Question : What is the value of $\left[\frac{12}{(\sqrt5+\sqrt3)}+\frac{18}{(\sqrt{5}-\sqrt3)}\right]$?
Option 1: $15(\sqrt5–\sqrt3)$
Option 2: $3(5\sqrt5+\sqrt3)$
Option 3: $15(\sqrt5+\sqrt3)$
Option 4: $3(3\sqrt5+\sqrt3)$
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Correct Answer: $3(5\sqrt5+\sqrt3)$
Solution :
Given:
$\left[\frac{12}{(\sqrt5+\sqrt3)}+\frac{18}{(\sqrt{5}-\sqrt3)}\right]$
= $\frac{12\sqrt5-12\sqrt 3+18\sqrt5+18\sqrt 3}{(\sqrt5+\sqrt3)(\sqrt{5}-\sqrt3)}$
= $\frac{30\sqrt5+6\sqrt3}{5-3}$
= $\frac{6(5\sqrt5+\sqrt3)}{2}$
= $3(5\sqrt5+\sqrt3)$
Hence, the correct answer is $3(5\sqrt5+\sqrt3)$.
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