If $\omega = \frac{-1 + \sqrt{3}i}{2}$,then $(3 + \omega + 3\omega^2)^4 = $

If $\omega$ is a complex cube root of unity,then the value of $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}$ is:

One of the $15^{\text{th}}$ roots of $-1$ is

Which of the following is a fourth root of $\frac{1}{2} + \frac{i\sqrt{3}}{2}$?

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

The smallest positive integral value of $n$ such that $\left[\frac{1+\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}}{1+\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}}\right]^{n} = 1$ is: