If ω is the cube root of unity, then find the value of ((-1 + i√3)/2)^18 + ((-1 - i√3)/2)^18

Question

If ω is the cube root of unity, then find the value of \((\frac{-1+i\sqrt3}2)^{18}+(\frac{-1-i\sqrt3}2)^{18}\)

((-1 + i√3)/2)¹⁸ + ((-1 - i√3)/2)¹⁸

Answer 1

If ω is the complex cube root of unity, then

Given Expression = ω¹⁸ + (ω²)¹⁸

= ω¹⁸ + ω³⁶

= (ω³)⁶ + (ω³)¹²

= (1)⁶ + (1)¹²

= 2