Question

Find the value of :

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

Answer 1

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

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

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

$=5\sqrt{3}\left|\begin{array}{ccc}1& 2& 1\\ \sqrt{5}& \sqrt{5}& \sqrt{2}\\ \sqrt{3}& \sqrt{3}& \sqrt{5}\end{array}\right|$

$=5\sqrt{3}(1\left|\begin{array}{cc}\sqrt{5}& \sqrt{2}\\ \sqrt{3}& \sqrt{5}\end{array}\right|-2\left|\begin{array}{cc}\sqrt{5}& \sqrt{2}\\ \sqrt{3}& \sqrt{5}\end{array}\right|+1\left|\begin{array}{cc}\sqrt{5}& \sqrt{5}\\ \sqrt{3}& \sqrt{3}\end{array}\right|)$

$=5\sqrt{3}\left(\right(5-\sqrt{6})-2(5-\sqrt{6}\left)\right)$

$=5\sqrt{3}(-5+\sqrt{6})$

$=15\sqrt{2}-25\sqrt{3}$