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

One of the values of $(\sqrt{3}-i)^{\frac{2}{5}}$ is

When $n=8$,$(\sqrt{3}+i)^n+(\sqrt{3}-i)^n=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 27 = 0$,find the quadratic equation whose roots are $\left( \frac{\gamma}{\alpha} \right)^2$ and $\left( \frac{\beta}{\alpha} \right)^2$.

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6}x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a, b$ be integers not divisible by $3$ and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+ib)$,where $i=\sqrt{-1}$. Then $n+a+b$ is equal to:

Two values of $(-8-8 \sqrt{3} i)^{1/4}$ are