Question

$\text{Find the value of}\sqrt{248+\sqrt{52+\sqrt{144}}}.$

• A. 13
• B. 12
• C. 14
• D. 16

Answer 1

The correct option is D 16

$\text{To find: Value of}\sqrt{248+\sqrt{52+\sqrt{144}}}$

$\text{Prime factorisation of 144 is:}$
$\begin{array}{cc}2& 144\\ 2& 72\\ 2& 36\\ 2& 18\\ 3& 9\\ 3& 3\\ & 1\end{array}$

$=\sqrt{248+\sqrt{52+\sqrt{2\times 2\times 2\times 2\times 3\times 3}}}$
$=\sqrt{248+\sqrt{52+(2\times 2\times 3)}}$
$=\sqrt{248+\sqrt{52+12}}=\sqrt{248+\sqrt{64}}$

$\text{Prime factorisation of 64 is:}$
$\begin{array}{cc}2& 64\\ 2& 32\\ 2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$

$=\sqrt{248+\sqrt{2\times 2\times 2\times 2\times 2\times 2}}$
$=\sqrt{248+(2\times 2\times 2)}$
$=\sqrt{248+8}=\sqrt{256}$

$\text{Prime factorisation of 256 is:}$
$\begin{array}{cc}2& 256\\ 2& 128\\ 2& 64\\ 2& 32\\ 2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$

$=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}$
$=2\times 2\times 2\times 2=16$