If $(1 + i\sqrt{3})^9 = a + ib$,then $b$ is equal to

If $\omega_0, \omega_1, \ldots, \omega_{n-1}$ are the $n$-th roots of unity,then $(1+2 \omega_0)(1+2 \omega_1)(1+2 \omega_2) \ldots (1+2 \omega_{n-1})=$

$\left(\frac{1+\cos \frac{\pi}{8}-i \sin \frac{\pi}{8}}{1+\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}}\right)^{12} = $

If $\omega$ is a complex root of the equation $z^3 = 1$,then $\omega + \omega^{\left( \frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \frac{27}{128} + \dots \right)}$ is equal to

If $1, \omega, \omega^2$ are the cube roots of unity,then $\omega^2(1 + \omega)^3 - (1 + \omega^2)\omega = $

If $\omega$ is a non-real root of the equation $x^3 - 1 = 0$,then the value of $\sum_{r=1}^{5} (1 + \omega^r + \omega^{2r})$ is