De Moivre's Theorem

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

The value of ${\left(\frac{1+\sin \frac{2\mathrm{\pi }}{9}+i\cos \frac{2\mathrm{\pi }}{9}}{1+\sin \frac{2\mathrm{\pi }}{9}-i\cos \frac{2\mathrm{\pi }}{9}}\right)}^3$ is

If $(\sqrt{3}+i{)}^{100}=2^{99}(p+iq),$ then $p$ and $q$ are roots of the equation :

The value of${\left(\frac{-1+i\sqrt{3}}{1-i}\right)}^{30}$ is :

The value of ${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$ is 

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2x+2=0,$ then ${\alpha }^{15}+{\beta }^{15}$ is equal to

If $z$ is a non-real complex number, then the minimum value of $\frac{Im z^5}{{\left(Im z\right)}^5}$ is (Where $Im z =$ Imaginary part of $z$)