The product $(1 - \omega + {\omega ^2})(1 - {\omega ^2} + {\omega ^4})(1 - {\omega ^4} + {\omega ^8}) \dots$ up to $2n$ factors is:

The least positive integer $n$ for which $\left( \frac{1 + i\sqrt{3}}{1 - i\sqrt{3}} \right)^n = 1$ is?

If $z^2+z+1=0$,then find the value of $\left(z^3+\frac{1}{z^3}\right)^2+\left(z^4+\frac{1}{z^4}\right)^2$,where $z$ is a complex cube root of unity.

If $\omega$ is a cube root of unity but not equal to $1$,then the minimum value of $|a + b\omega + c\omega^2|$ (where $a, b, c$ are integers but not all equal) is

The $n^{th}$ roots of unity are in

If $|Z|=2$, $Z_1=\frac{Z}{2} e^{i \alpha}$ and $\theta$ is the $\operatorname{amp}(Z)$, then $\frac{Z_1^n-Z_1^{-n}}{Z_1^n+Z_1^{-n}}=$