Answer
$52+6\sqrt{35}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(3\sqrt{5}+2\sqrt{7})^2
,$ use the special product on squaring binomials and the laws of radicals.
$\bf{\text{Solution Details:}}$
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 expression above is equivalent to
\begin{array}{l}\require{cancel}
(3\sqrt{5})^2+2(3\sqrt{5})(\sqrt{7})+(\sqrt{7})^2
\\\\=
9(5)+6(\sqrt{5})(\sqrt{7})+7
\\\\=
45+6(\sqrt{5})(\sqrt{7})+7
\\\\=
52+6(\sqrt{5})(\sqrt{7})
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
52+6\sqrt{5(7)}
\\\\=
52+6\sqrt{35}
.\end{array}
