$(1-i \sqrt{3})^{2025}=$

If $\alpha, \beta$ are the roots of the equation $x+\frac{4}{x}=2 \sqrt{3}$,then $\frac{2}{\sqrt{3}}\left|\alpha^{2024}-\beta^{2024}\right|=$

If $\text{cis } \alpha$ is the common value of $(-1)^{1/4}$ and $(-i)^{1/2}$,then $\tan \alpha = $

The value of $\sum\limits_{k = 1}^6 {\left( {\sin \frac{{2\pi k}}{7} - i\cos \frac{{2\pi k}}{7}} \right)} $ is

The sum of the complex roots of the equation $(x-1)^3+64=0$ is

Let $p, q \in \mathbb{R}$ and $(1-\sqrt{3}i)^{200} = 2^{199}(p + iq)$,where $i = \sqrt{-1}$. Then $p + q + q^2$ and $p - q + q^2$ are roots of the equation: