Question

Answer 1

Ta có: \(x=\sqrt{3+\sqrt{5+2\sqrt3}}+\sqrt{3-\sqrt{5+2\sqrt3}}\)

=>\(x^2=3+\sqrt{5+2\sqrt3}+3-\sqrt{5+2\sqrt3}+2\cdot\sqrt{\left(3+\sqrt{5+2\sqrt3}\right)\left(3-\sqrt{5+2\sqrt3}\right)}\)

=>\(x^2=6+2\cdot\sqrt{9-5-2\sqrt3}=6+2\cdot\sqrt{4-2\sqrt3}=6+2\cdot\sqrt{\left(\sqrt3-1\right)^2}\)

=>\(x^2=6+2\left(\sqrt3-1\right)=6+2\sqrt3-2=4+2\sqrt3=\left(\sqrt3+1\right)^2\)

=>\(x=\sqrt3+1\)

\(Q=\left(x^2-2x-1\right)^{2021}\)

\(=\left(4+2\sqrt3-2\sqrt3-2-1\right)^{2021}=1^{2021}=1\)