What is the product of all the values of $(1-i)^{\frac{2}{5}}$?

If $\theta = \frac{\pi}{6}$,then the $10^{th}$ term of the series $1 + (\cos \theta + i \sin \theta) + (\cos \theta + i \sin \theta)^2 + (\cos \theta + i \sin \theta)^3 + \ldots$ is equal to:

If ${z_1}, {z_2}, {z_3}, ......, {z_n}$ are the $n^{th}$ roots of unity,then for $k = 1, 2, ....., n-1$:

If $\alpha, \beta$ are the roots of $1+x+x^2=0$,then the value of $\alpha^4+\beta^4+\alpha^{-4}\beta^{-4}$ is

If $(\sqrt{3}+i)^8-(\sqrt{3}-i)^8=\alpha+i \beta$,then $\alpha-\frac{\sqrt{3}}{2} \beta=$

The square of either of the two imaginary cube roots of unity is: