\(\sqrt{6+2\sqrt{5-\sqrt{13+4\sqrt{3}}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{12+2\cdot2\sqrt{3}+1}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{\left(\sqrt{12}+1\right)^2}}}\)
\(=\sqrt{6+2\sqrt{5-\left|2\sqrt{3}+1\right|}}\)
\(=\sqrt{6+2\sqrt{5-\left(2\sqrt{3}+1\right)}}\) (vì \(2\sqrt{3}+1>0\))
\(=\sqrt{6+2\sqrt{5-2\sqrt{3}-1}}\\ =\sqrt{6+2\sqrt{4-2\sqrt{3}}}\\ =\sqrt{6+2\sqrt{3-2\cdot\sqrt{3}+1}}\\ =\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\sqrt{6+2\cdot\left|\sqrt{3}-1\right|}\)
\(=\sqrt{6+2\cdot\left(\sqrt{3}-1\right)}\) (vì \(\sqrt{3}-1>0\))
\(=\sqrt{6+2\sqrt{3}-2}\\ =\sqrt{4+2\sqrt{3}}\\ =\sqrt{3+2\cdot\sqrt{3}+1}\\ =\sqrt{\left(\sqrt{3}+1\right)^2}\\ =\left|\sqrt{3}+1\right|\)
\(=\sqrt{3}+1\) (vì \(\sqrt{3}+1>0\))
