If $2 \alpha = -1 - i \sqrt{3}$ and $2 \beta = -1 + i \sqrt{3}$,then $5 \alpha^4 + 5 \beta^4 + 7 \alpha^{-1} \beta^{-1}$ is equal to

If $a=\cos \left(\frac{8 \pi}{11}\right)+i \sin \left(\frac{8 \pi}{11}\right)$,then $\operatorname{Re}\left(a+a^2+a^3+a^4+a^5\right)=$

If $\omega$ is a non-real cube root of unity,then $(a + b)(a + b\omega)(a + b\omega^2)$ is equal to:

Let $\alpha = \frac{-1 + i \sqrt{3}}{2}$. If $a = (1 + \alpha) \sum_{k=0}^{100} \alpha^{2k}$ and $b = \sum_{k=0}^{100} \alpha^{3k}$,then $a$ and $b$ are the roots of the quadratic equation:

If $\omega$ is a cube root of unity,then $(1 + \omega)^3 - (1 + \omega^2)^3 = $

If $\omega$ is a complex cube root of unity, then the value of the expression $2(1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}) + 3(2 + \frac{1}{\omega})(2 + \frac{1}{\omega^2}) + ... + (n + 1)(n + \frac{1}{\omega})(n + \frac{1}{\omega^2})$ is...