${\left( \frac{\sqrt{3} + i}{2} \right)^6} + {\left( \frac{i - \sqrt{3}}{2} \right)^6}$ is equal to

For the equation $x^2 + x + 1 = 0$,if $\alpha$ and $\beta$ are the roots,then which of the following equations has roots $\alpha^{19}$ and $\beta^{7}$?

If $\alpha, \beta$ are the roots of $x^2-2x+4=0$, for $n \in N$, what is the value of $\alpha^n+\beta^n$?

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^{4}+x^{3}+x^{2}+x+1=0$,then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to

If $(\sqrt{3}-i)^{n}=2^{n}, n \in N$,then the least possible value of $n$ is

Let $z = \frac{1 - i \sqrt{3}}{2}$,where $i = \sqrt{-1}$. Then the value of $21 + \left(z + \frac{1}{z}\right)^{3} + \left(z^{2} + \frac{1}{z^{2}}\right)^{3} + \left(z^{3} + \frac{1}{z^{3}}\right)^{3} + \dots + \left(z^{21} + \frac{1}{z^{21}}\right)^{3}$ is .... .