If $\alpha =\frac{-1+\sqrt{-3}}{2}$ then what is the value of $(1+\alpha ^{19}-\alpha ^{35})^{100}-(1-3\alpha ^{25}+\alpha ^{38})^{50}? $
 

If  $(\frac{1-i}{1+i})^{2m}(\frac{1+i}{1-i})^{2n}=1$where $i=\sqrt{-1}$ then what is the smallest positive value of (m - n)?

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?