Đặt \(x=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}>0\)
\(x^2=6+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}\)
\(\Rightarrow x^2=6+2\sqrt{4-2\sqrt{3}}\)
\(\Rightarrow x^2=6+2\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(\Rightarrow x^2=6+2\left(\sqrt{3}-1\right)=4+2\sqrt{3}\)
\(\Rightarrow x^2=\left(\sqrt{3}+1\right)^2\)
\(\Rightarrow x=\sqrt{3}+1\)
\(A=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\)
\(A^2=3+\sqrt{5+2\sqrt{3}}+3-\sqrt{5+2\sqrt{3}}+2\sqrt{9-\left(5+2\sqrt{3}\right)}\)
\(=6+2\sqrt{4-2\sqrt{3}}\)
\(=6+2\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=6+2\sqrt{3}-2\)
\(=\left(\sqrt{3}+1\right)^2\)
\(\Rightarrow A=\sqrt{3}+1\)
