If $\omega$ is a complex cube root of unity,then the value of $\omega^{99} + \omega^{100} + \omega^{101}$ is

One of the roots of the equation $(x+1)^4 + 81 = 0$ is

Let $\alpha$ and $\beta$ be the roots of $x^2 + \omega x + \omega^2 = 0$,where $\omega$ is an imaginary cube root of unity. If $z = \alpha^9 + i\beta^9$,then the value of $|z|$ is:

$n$ is a positive integer and not a multiple of $3$. If $\omega$ is a non-real cube root of unity,then $\omega^n + \omega^{2n}$ is equal to

If $\omega$ is a complex cube root of unity,then $\omega^{\left(\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\ldots \infty\right)}+\omega^{\left(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\ldots \infty\right)}$ is equal to

Let $z = \left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)^5 + \left( \frac{\sqrt{3}}{2} - \frac{i}{2} \right)^5$. If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z$,then: