The real part of $z$ that satisfies $i z^4+1=0$ is

If $(\sqrt{3}+i)^8-(\sqrt{3}-i)^8=\alpha+i \beta$,then $\alpha-\frac{\sqrt{3}}{2} \beta=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^9+\beta^9$ is equal to

If $\omega$ is a complex cube root of unity,then $(1 - \omega )(1 - {\omega ^2})(1 - {\omega ^4})(1 - {\omega ^8}) = $

If $n$ is a positive integer, then $(1 + i\sqrt{3})^{2n} + (1 - i\sqrt{3})^{2n}$ 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 $.......$