Question

find the square root of
$9 + 2 \sqrt{15} + 2 \sqrt{5} + 2 \sqrt{3}$
solve it please
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Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

We are aware of the identity that

$(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca$

QUESTION

TO DETERMINE

$9 + 2 \sqrt{15} + 2 \sqrt{5} + 2 \sqrt{3}$

CALCULATION

$9 + 2 \sqrt{15} + 2 \sqrt{5} + 2 \sqrt{3}$

$= 5 + 3 + 1 + 2 \sqrt{15} + 2 \sqrt{5} + 2 \sqrt{3}$

$= {( \sqrt{5}) }^{2} + {( \sqrt{3} )}^{2} + {1}^{2} + 2 \sqrt{5} \times \sqrt{3} + 2 \sqrt{5} \times \sqrt{1} + 2 \sqrt{3} \times \sqrt{1}$

$= {( \sqrt{5} + \sqrt{3} + 1) }^{2} \: \: \: using \: \: the \: above \: identity$

Hence

$\sqrt{9 + 2 \sqrt{15} + 2 \sqrt{5} + 2 \sqrt{3}}$

$= \sqrt{5} + \sqrt{3} + 1$