x²+x+1=0

\(\Leftrightarrow\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{3}{4}=0\)

\(\Leftrightarrow\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}=0\)

\(\Rightarrow x\in\varnothing\)

\(\Rightarrow A=x^n+\dfrac{1}{x^n}\in\varnothing\)

x^​2​+x+1\(\ne\)0 with any value of x

So don't have any value of x satisfy

So don't have any value of A

So A\(\in\)\(\varnothing\)