Question

\(2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}\)

Answer 1

Ta có : \(2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}\) \(=2\sqrt{3+\sqrt{5-\sqrt{12+2\sqrt{12}+1}}}\)

\(=2\sqrt{3+\sqrt{5-\sqrt{\left(\sqrt{12}+1\right)^2}}}\)\(=2\sqrt{3+\sqrt{5-\sqrt{12}-1}}\)

\(=2\sqrt{3+\sqrt{3-2\sqrt{3}+1}}\)\(=2\sqrt{3+\sqrt{\left(\sqrt{3}-1\right)^2}}=2\sqrt{3+\sqrt{3}-1}\)

\(=2\sqrt{2+\sqrt{3}}=\sqrt{2}\sqrt{4+2\sqrt{3}}=\sqrt{2}\sqrt{\left(\sqrt{3}+1\right)^2}=\sqrt{2}\left(\sqrt{3}+1\right)\)