Question

Rút gọn biểu thức sau :

\(N=\sqrt[4]{2+\sqrt{5}+2\sqrt{2+\sqrt{5}}}+\sqrt[4]{2+\sqrt{5}-2\sqrt{2+\sqrt{5}}}\)

Answer 1

Đặt \(A=\sqrt[4]{2+\sqrt{5}+2\sqrt{2+\sqrt{5}}};B=\sqrt[4]{2+\sqrt{5}-2\sqrt{2+\sqrt{5}}}\)

\(\Rightarrow N=A+B\)

Ta có \(AB=\sqrt[4]{\left(2+\sqrt{5}\right)^2-4\left(\sqrt{2+\sqrt{5}}\right)}=1\)

và \(A^4+B^4=4+2\sqrt{5}\)

Suy ra \(A^4+B^4=2A^2B^2=6+2\sqrt{5}=\left(\sqrt{5}+1\right)^2\)

          \(\Leftrightarrow A^2+B^2=\sqrt{5}+1\)

Tức là :

\(A^2+B^2+2AB=\sqrt{5}+3=\left(\frac{\sqrt{5}+1}{\sqrt{2}}\right)^2\)

\(\Leftrightarrow A+B=\frac{\sqrt{5}+1}{\sqrt{2}}\)

Vậy \(N=\frac{\sqrt{5}+1}{\sqrt{2}}\)