If $\sqrt{x}+\frac{1}{\sqrt{x}}=2 \cos \theta$,then $x^6+x^{-6}=$

If $z_1$ and $z_2$ are two of the $n^{\text{th}}$ roots of unity such that the line segment joining them subtends a right angle at the origin,then for a positive integer $k$,$n$ takes the form

If $z=\frac{\sqrt{3}+i}{2}$,then $\left(z^{101}+i^{103}\right)^{105}=$

The value of $\sum\limits_{r = 1}^8 {\left( {\sin \frac{{2r\pi }}{9} + i\cos \frac{{2r\pi }}{9}} \right)} $ is

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

If $1, \omega, \omega^2, \ldots, \omega^{10}$ are the $11^{th}$ roots of unity,then the product of these roots is: