Question

\(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)

Answer 1

\(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)

\(=\sqrt{2+2\sqrt{2}.\sqrt{3}+3}-\sqrt{2-2.\sqrt{2}.\sqrt{3}+3}\)

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

\(=\sqrt{2}+\sqrt{3}-\left|\sqrt{3}-\sqrt{2}\right|\)

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

\(=2\sqrt{2}\).