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

If $\omega$ is a complex cube root of unity,then $(x+1)(x+\omega)(x-\omega-1)$ is equal to

If the cube roots of unity are $1, \omega, \omega^2$,then the roots of the equation $(x - 1)^3 + 8 = 0$ are

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 $z=$

Consider a regular $10$-gon with its vertices on the unit circle. With one vertex fixed,draw straight lines to the other $9$ vertices. Call them $L_1, L_2, \ldots, L_9$ and denote their lengths by $l_1, l_2, \ldots, l_9$ respectively. Then,the product $l_1 \times l_2 \times \ldots \times l_9$ is

Let $\omega$ be an imaginary cube root of unity. Then the value of $2(\omega + 1)(\omega^2 + 1) + 3(2\omega + 1)(2\omega^2 + 1) + \dots + (n + 1)(n\omega + 1)(n\omega^2 + 1)$ is