Question : What is the value of $\frac{\sqrt{24} + \sqrt{216}}{\sqrt{96}}$?
Option 1: $2\sqrt{6}$
Option 2: $4\sqrt{6}$
Option 3: $2$
Option 4: $4$
Correct Answer: $2$
Solution :
Given: $\frac{\sqrt{24} + \sqrt{216}}{\sqrt{96}}$
By factorising and simplifying this expression,
$\frac{\sqrt{2\times2\times2\times 3} + \sqrt{2\times2\times2\times3\times3\times3}}{\sqrt{2\times2\times2\times2\times2\times3}}$
= $\frac{\sqrt{4\times6} + \sqrt{6\times36}}{\sqrt{6\times16}}$
= $\frac{2\sqrt{6} + 6\sqrt{6}}{4\sqrt{6}}$
= $\frac{(2+6)\sqrt{6}}{4\sqrt{6}}$
= $\frac{8\sqrt{6}}{4\sqrt{6}}$
= $\frac{8}{4}$
= $2$
Hence, the correct answer is $2$.
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