If $w = \frac{-1 + i \sqrt{3}}{2}$,where $i = \sqrt{-1}$,then the value of $(3 + w + 3 w^2)^4$ is

If $\alpha, \beta$ are the roots of the equation $x^2+x+1=0$,then $(\alpha+\beta)^2+(\alpha^2+\beta^2)^2+(\alpha^3+\beta^3)^2+\ldots+(\alpha^{12}+\beta^{12})^2=$

$\frac{(\sin \frac{\pi}{8} + i \cos \frac{\pi}{8})^8}{(\sin \frac{\pi}{8} - i \cos \frac{\pi}{8})^8}$ is equal to

If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 = 1$,then the value of $\sum_{i=1}^4 z_i^3$ is

If $z = \cos \theta + i \sin \theta$,then $z^r + (\bar{z})^r = $

If $x_n = \cos \frac{\pi}{2^n} + i \sin \frac{\pi}{2^n}$,then $\prod_{n=1}^{\infty} x_n$ is equal to