The product of all the roots of ${\left( \cos \frac{\pi }{3} + i\sin \frac{\pi }{3} \right)^{3/4}}$ is

If $\omega$ is a non-real root of the equation $x^3 - 1 = 0$,then the value of $\sum_{r=1}^{5} (1 + \omega^r + \omega^{2r})$ is

$(\sqrt{\sqrt{2}+1} + i\sqrt{\sqrt{2}-1})^8 =$

If $1, \omega, \omega^2$ are the cube roots of unity,then the value of $(x+y)^2+(x \omega+y \omega^2)^2+(x \omega^2+y \omega)^2$ is

Let $\alpha, \beta$ denote the cube roots of unity other than $1$ and $\alpha \neq \beta$. Let $S = \sum_{n=0}^{\infty} (-1)^{n} \left(\frac{\alpha}{\beta}\right)^{n}$. Then the value of $S$ is

If $1, \omega, \omega^2$ are the cube roots of unity and $1, \alpha, \alpha^2, \alpha^3$ are the fourth roots of unity in usual notation,then $\alpha+\alpha \omega-\alpha^3 \omega^2=$