The value of $(\frac{-1 + i\sqrt{3}}{2})^{18} + (\frac{-1 - i\sqrt{3}}{2})^{18}$ 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 $\alpha+\alpha \omega-\alpha^3 \omega^2=$

The cube roots of unity when represented on the Argand plane form the vertices of an

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}$ and $\beta^7$ is

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

If $\omega$ is a complex cube root of unity,then $\cos \left[\left(\omega^{1234}+\omega^{2021}\right) \pi-\frac{\pi}{4}\right]$ is equal to