Question

Tính:

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

Answer 1

\(E=\sqrt{6+2\sqrt{5-\sqrt{13+\sqrt{48}}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{13+\sqrt{16\cdot3}}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{13+4\sqrt{3}}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{12+2\cdot2\sqrt{3}\cdot1+1}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{\left(2\sqrt{3}+1\right)^2}}}\\ =\sqrt{6+2\sqrt{5-2\sqrt{3}-1}}\\ =\sqrt{6+2\sqrt{4-2\sqrt{3}}}\\ =\sqrt{6+2\sqrt{3-2\cdot\sqrt{3}\cdot1+1}}\\ =\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}\\ =\sqrt{6+2\left(\sqrt{3}-1\right)}\\ =\sqrt{6+2\sqrt{3}-2}\\ =\sqrt{4+2\sqrt{3}}\\ \)

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