If $1, \omega, \omega^2$ are the cube roots of unity,then their product is

If $\omega \neq 1$ is a cube root of unity, then the sum of the series $S = 1 + 2\omega + 3\omega^2 + \dots + 3n\omega^{3n-1}$ is

One of the values of $(-32 i)^{\frac{2}{5}}$ is

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2+x+1=0$,then the equation whose roots are $\alpha^{2021}$ and $\beta^{2021}$ is given by $.......$

One of the roots of the equation $x^{14}+x^9-x^5-1=0$ is

If $x+\frac{1}{x}=2 \cos \theta$, then for any integer $n$, $x^n+\frac{1}{x^n}=$