The product of all the values of $(\sqrt{3}-i)^{2/5}$ is

Let $i=\sqrt{-1}$. If $\frac{(-1+i \sqrt{3})^{21}}{(1-i)^{24}}+\frac{(1+i \sqrt{3})^{21}}{(1+i)^{24}}=k$,and $n =[| k |]$ be the greatest integral part of $| k |$. Then $\sum_{ j =0}^{ n +5}( j +5)^{2}-\sum_{ j =0}^{ n +5}( j +5)$ is equal to ........ .

If $\omega (\neq 1)$ is a cube root of unity and $(1 + \omega)^7 = A + B\omega$,then $A$ and $B$ are equal to:

Let $z_1, z_2, \ldots, z_7$ be the vertices of a regular heptagon inscribed in the unit circle with the center at the origin in the complex plane. If $w = \sum_{1 \leq i < j \leq 7} z_i z_j$,then $|w|$ is equal to:

If $\omega$ is a complex cube root of unity,then the value of $\omega^{99} + \omega^{100} + \omega^{101}$ is

If the complex number $a$ is such that $|a|=1$ and $\arg (a)=\theta$,then the roots of the equation $\left(\frac{1+i z}{1-i z}\right)^4=a$ are $z=$