Question

\(\sqrt{6+2\sqrt{5-\sqrt{13+\sqrt{48}}}}\)

Answer 1

Ta có : \(\sqrt{6+2\sqrt{5-\sqrt{13+\sqrt{48}}}}\)\(=\sqrt{6+2\sqrt{5-\sqrt{12+2\sqrt{12}+1}}}\)

\(=\sqrt{6+2\sqrt{5-\sqrt{\left(\sqrt{12}+1\right)^2}}}\)\(=\sqrt{6+2\sqrt{4-\sqrt{12}}}=\sqrt{6+2\sqrt{3-2\sqrt{3}+1}}\)

\(=\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}=\sqrt{6+2\left(\sqrt{3}-1\right)}=\sqrt{6+2\sqrt{3}-2}\)

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