Question : Simplify the following.
$\frac{\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}}{\sqrt{16+19.25 \times 4^2}}$
Option 1: $\frac{7}{18}$
Option 2: $\frac{2}{9}$
Option 3: $\frac{5}{18}$
Option 4: $\frac{1}{9}$
Correct Answer: $\frac{2}{9}$
Solution :
$\frac{\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}}{\sqrt{16+19.25 \times 4^2}}$
$=\frac{\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+15}}}}}{\sqrt{4^2(1+19.25)}}$
$=\frac{\sqrt{10+\sqrt{25+\sqrt{108+13}}}}{4\sqrt{20.25}}$
$=\frac{\sqrt{10+\sqrt{25+11}}}{4\ \times 4.5}$
$=\frac{\sqrt{10+6}}{18}$
$=\frac{4}{18}$
$=\frac{2}{9}$
Hence, the correct answer is $\frac{2}{9}$.
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