One of the values of $(-32 i)^{\frac{2}{5}}$ is

If the least positive integer $n$ satisfying the equation $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^{n}=-1$ is $p$ and the least positive integer $m$ satisfying the equation $\left(\frac{1-\sqrt{3} i}{1+\sqrt{3} i}\right)^m=\operatorname{cis} \frac{2 \pi}{3}$ is $q$,then $\sqrt{p^2+q^2}=$

$\left(\frac{1+\cos \frac{\pi}{8}-i \sin \frac{\pi}{8}}{1+\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}}\right)^8$ is equal to

If $Z = r(\cos \theta + i \sin \theta), (\theta \neq -\pi / 2)$ is a solution of $x^3 = i$,then $r^9(\cos \theta + i \sin \theta)^9 = $

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

$n$ is a positive integer and not a multiple of $3$. If $\omega$ is a non-real cube root of unity,then $\omega^n + \omega^{2n}$ is equal to