Question

Find $(5\sqrt{3}+3\sqrt{2})\Delta$ where
$\Delta =\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& \sqrt{5}& 2\sqrt{5}\\ \sqrt{15}+\sqrt{26}& \sqrt{10}& 5\\ 3+\sqrt{65}& 5& \sqrt{15}\end{array}\right|$

Answer 1

$\Delta =\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& \sqrt{5}& 2\sqrt{5}\\ \sqrt{15}+\sqrt{26}& \sqrt{10}& 5\\ 3+\sqrt{65}& 5& \sqrt{15}\end{array}\right|$

$\Rightarrow \Delta =\sqrt{5}\sqrt{5}\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& 1& 2\\ \sqrt{15}+\sqrt{26}& \sqrt{2}& \sqrt{5}\\ 3+\sqrt{65}& \sqrt{5}& \sqrt{3}\end{array}\right|=5\left|\begin{array}{ccc}\sqrt{13}& 1& 2\\ \sqrt{26}& \sqrt{2}& \sqrt{5}\\ \sqrt{65}& \sqrt{5}& \sqrt{3}\end{array}\right|+5\sqrt{3}\left|\begin{array}{ccc}1& 1& 2\\ \sqrt{5}& \sqrt{2}& \sqrt{5}\\ \sqrt{3}& \sqrt{5}& \sqrt{3}\end{array}\right|$

$\Rightarrow \Delta =5\sqrt{3}(5-\sqrt{6})$
$\therefore (5\sqrt{3}+3\sqrt{2})\Delta =285$