Answer
$84+24\sqrt{6}$
Work Step by Step
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the given expression, $
\left( \sqrt{12}+\sqrt{72} \right)^2
,$ is equivalent to
\begin{align*}
&
\left( \sqrt{12}\right)^2+2\left( \sqrt{12}\right)\left(\sqrt{72} \right)+\left(\sqrt{72} \right)^2
\\\\&=
12+2\left( \sqrt{12(72)}\right)+72
\\\\&=
(12+72)+2\sqrt{12(12\cdot6)}
\\\\&=
84+2\sqrt{12^2\cdot6}
\\\\&=
84+2(12)\sqrt{6}
\\\\&=
84+24\sqrt{6}
.\end{align*}
Hence, the simplified form of the given expression is $
84+24\sqrt{6}
$.
