Question

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

Answer 1

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

\(=\sqrt{2+\sqrt{5-\sqrt{13+4\sqrt{3}}}}\)

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

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

\(=\sqrt{2+\sqrt{5-1-2\sqrt{3}}}\)

\(=\sqrt{2+\sqrt{4-2\sqrt{3}}}\)

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

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

\(=\sqrt{1+\sqrt{3}}\)

Answer 2

\(\sqrt{2+\sqrt{5-\sqrt{13+\sqrt{48}}}}=\sqrt{2+\sqrt{5-\sqrt{12+4\sqrt{3}+1}}}\)

\(=\sqrt{2+\sqrt{4-2\sqrt{3}}}=\sqrt{1+\sqrt{3}}\)