What is the value of $(1-i \sqrt{3})^9$?

If $1, \omega, \omega^2$ are the three cube roots of unity,then $(3 + \omega^2 + \omega^4)^6 = $

If $Z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right)$ for $r = 1, 2, 3, \ldots$,then the product $Z_1 Z_2 Z_3 \ldots \infty$ is equal to:

If $\alpha$ is a non-real root of $x^6=1$,then $\frac{\alpha^5+\alpha^3+\alpha+1}{\alpha^2+1}$ is equal to

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2+x+1=0$,then the equation whose roots are $\alpha^{2021}$ and $\beta^{2021}$ is given by $.......$

For $n \in Z^{+}$,$(1+\sin \theta+i \cos \theta)^n+(1+\sin \theta-i \cos \theta)^n=$