$(27)^{1/3} = $

If $\alpha, \beta$ are the roots of the equation $1+x+x^2=0$,then $(2-\alpha)(2-\beta)(2-\alpha^{10})(2-\alpha^{20})=$

If $\sqrt{x}+\frac{1}{\sqrt{x}}=2 \cos \theta$,then $x^6+x^{-6}=$

One of the values of $(-64 i)^{5 / 6}$ is

If $\cos \theta + i \sin \theta, \theta \in R$, is a root of the equation $a_0 x^n + a_1 x^{n-1} + \ldots + a_{n-1} x + a_n = 0$, where $a_0, a_1, \ldots, a_n \in R$ and $a_0 \neq 0$, then the value of $a_1 \sin \theta + a_2 \sin 2 \theta + \ldots + a_n \sin n \theta$ is:

Let $\alpha, \beta$ denote the cube roots of unity other than $1$ and $\alpha \neq \beta$. Let $S = \sum_{n=0}^{\infty} (-1)^{n} \left(\frac{\alpha}{\beta}\right)^{n}$. Then the value of $S$ is