Question

A=\(\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{4+2\sqrt{3}}}}\)

Answer 1

\(A=\sqrt{6+2\sqrt{2}\cdot\sqrt{3-\sqrt{4+2\sqrt{3}}}}\\ =\sqrt{6+2\sqrt{2}\cdot\sqrt{3-\sqrt{3+2\cdot\sqrt{3}\cdot1+1}}}\\ =\sqrt{6+2\sqrt{2}\cdot\sqrt{3-\sqrt{\left(\sqrt{3}+1\right)^2}}}\\ =\sqrt{6+2\sqrt{2}\cdot\sqrt{2-\sqrt{3}}}\\ =\sqrt{6+2\sqrt{4-2\sqrt{3}}}\\ =\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}\\ =\sqrt{6+2\left(\sqrt{3}-1\right)}\\ =\sqrt{4+2\sqrt{3}}\\ =\sqrt{3+2\cdot\sqrt{3}\cdot1+1}\\ =\sqrt{\left(\sqrt{3}+1\right)^2}=\sqrt{3}+1\)