\(\sqrt{\left(5+2\sqrt{6}\right)}+\sqrt{8-2\sqrt{15}}\)
\(=\sqrt{\left(2+2.\sqrt{2}.\sqrt{3}+3\right)}+\sqrt{3-2\sqrt{3}.\sqrt{5}+5}\)
\(=\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{3}+\sqrt{5}\right)^2}\)
\(=\sqrt{2}+\sqrt{3}+\sqrt{3}+\sqrt{5}\)
\(=\sqrt{2}+2\sqrt{3}+\sqrt{5}\)