Question

\(\sqrt{10}\sqrt{\left(\sqrt{2}+1\right)\left(\sqrt{3}+1\right)\left(\sqrt{6}+1\right)\left(5-2\sqrt{2}-\sqrt{3}\right)}\)

Answer 1

\(=\sqrt{10}\cdot\sqrt{\left(\sqrt{6}+1+\sqrt{2}+\sqrt{3}\right)\left(\sqrt{6}+1\right)\left(5-2\sqrt{2}-\sqrt{3}\right)}\)

\(=\sqrt{10}\cdot\sqrt{\left(7+2\sqrt{6}+2\sqrt{3}+\sqrt{2}+3\sqrt{2}+\sqrt{3}\right)\left(5-2\sqrt{2}-\sqrt{3}\right)}\)

\(=\sqrt{10}\cdot\sqrt{\left(7+2\sqrt{6}+3\sqrt{3}+4\sqrt{2}\right)\left(5-2\sqrt{2}-\sqrt{3}\right)}\)

\(=\sqrt{10}\cdot\sqrt{35-14\sqrt{2}-7\sqrt{3}+10\sqrt{6}-8\sqrt{3}-6\sqrt{2}+15\sqrt{3}-6\sqrt{6}-9+20\sqrt{2}-16-4\sqrt{6}}\)

=10