Let 1, ω, $\omega ^ 2$ be three cube roots of unity. If x = a + b $y=a\omega +b\omega ^{2}, z=a\omega ^{2}+b\omega$ then what is $ x ^ 2 + y ^ 2 + z ^ 2$ equal to?

What is $ \left( \frac{\sqrt{3}+i}{\sqrt{3}-i} \right)^3 $ equal to?

If $z\ne0$ is a complex number, then what is $amp(z)+amp(\bar{z})$ equal to?

If ω ≠ 1 is a cube root of unity, then what is (1 + ω - ω²)¹⁰⁰ + (1 - ω + ω²)¹⁰⁰ equal to?