Cho x=[tex]\frac{2}{\frac{1}{\sqrt{\sqrt{2}+1}-1}-\frac{1}{\sqrt{\sqrt{2}+1}+1}}[/tex]. Tính giá trị biểu thức [tex]B=\left ( 1-2x+x^{2}+x^{3}-x^{4} \right )^{2018}[/tex]
[tex]x=\frac{2}{\frac{1}{\sqrt{\sqrt{2}+1}-1}-\frac{1}{\sqrt{\sqrt{2}+1}+1}}=\frac{2}{\frac{\sqrt{\sqrt{2}+1}+1-(\sqrt{\sqrt{2}+1}-1)}{(\sqrt{\sqrt{2}+1}-1)(\sqrt{\sqrt{2}+1}+1)}}=\frac{2}{\frac{2}{\sqrt{2}+1-1}}=\frac{2}{\frac{2}{\sqrt{2}}}=\frac{2}{\sqrt{2}}=\sqrt{2}[/tex]
Từ đó [tex]1-2x+x^{2}+x^{3}-x^{4}=x^4+x^2+1+x(x^2-2)=2^2+2+1+\sqrt{2}(2-2)=4+2+1=7\Rightarrow B=7^{2018}[/tex]