Question

\(\left(5+2\sqrt{6}\right)\left(49-20\sqrt{6}\right)\sqrt{5-2\sqrt{6}}\)

Answer 1

\(=\left(2+2\sqrt{2}\cdot\sqrt{3}+3\right)\left(25-2\cdot5\cdot\sqrt{24}+24\right)\sqrt{3-2\sqrt{3}\sqrt{2}+2}\)

\(=\left(\sqrt{2}+\sqrt{3}\right)^2\left(5-\sqrt{24}\right)^2\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)

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

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

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

\(=9\sqrt{3}-11\sqrt{2}\)