If $1, \omega, \omega^2, \ldots, \omega^8$ are the roots of the equation $x^9-1=0$,then $\sum_{r=1}^8 \left(\omega^r\right)^{99} =$

Let $\omega \neq 1$ be a cube root of unity. Then the minimum value of the set $\{|a + b\omega + c\omega^2|^2 : a, b, c \text{ are distinct non-zero integers}\}$ is equal to:

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

If $x = a + b$,$y = a\omega + b\omega^2$,and $z = a\omega^2 + b\omega$,then the value of $x^3 + y^3 + z^3$ is equal to

If $z^2+z+1=0$,where $z$ is a complex number,then $\left(z+\frac{1}{z}\right)^3+\left(z^4+\frac{1}{z^4}\right)^3$ is equal to

If $\alpha$ is a root of $z^2-z+1=0$,then $\left(\alpha^{2014}+\frac{1}{\alpha^{2014}}\right)+\left(\alpha^{2015}+\frac{1}{\alpha^{2015}}\right)^2+\left(\alpha^{2016}+\frac{1}{\alpha^{2016}}\right)^3+\left(\alpha^{2017}+\frac{1}{\alpha^{2017}}\right)^4+\left(\alpha^{2018}+\frac{1}{\alpha^{2018}}\right)^5=$