Question

Tính :\(\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\) 

Answer 1

Đặt \(x=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}>0\)

\(x^2=6+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}\)

\(\Rightarrow x^2=6+2\sqrt{4-2\sqrt{3}}\)

\(\Rightarrow x^2=6+2\sqrt{\left(\sqrt{3}-1\right)^2}\)

\(\Rightarrow x^2=6+2\left(\sqrt{3}-1\right)=4+2\sqrt{3}\)

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

\(\Rightarrow x=\sqrt{3}+1\)

Answer 2

\(A=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\)

\(A^2=3+\sqrt{5+2\sqrt{3}}+3-\sqrt{5+2\sqrt{3}}+2\sqrt{9-\left(5+2\sqrt{3}\right)}\)

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

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

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

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

\(\Rightarrow A=\sqrt{3}+1\)