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

If $x=a+b$,$y=a \alpha+b \beta$,$z=a \beta+b \alpha$ and $\alpha, \beta$ are the complex cube roots of unity,then $x^3+y^3+z^3=$

Evaluate: $[\sqrt{2}(\cos 56^{\circ} 15^{\prime} + i \sin 56^{\circ} 15^{\prime})]^8$

If $\alpha, \beta$ are the roots of the quadratic equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}, \beta^7$ is

For a non-$0$ complex number $z$,let $\arg (z)$ denote the principal argument of $z$,with $-\pi < \arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg (\omega) < \pi$. Let $\alpha = \arg \left(\sum_{n=1}^{2025} (-\omega)^n\right)$. Then the value of $\frac{3 \alpha}{\pi}$ is $.....$ .

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