If $i = \sqrt{-1}$,then $4 + 5\left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right)^{334} + 3\left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right)^{365}$ is equal to

The product of the distinct $(2n)^{\text{th}}$ roots of $1+i\sqrt{3}$ is equal to:

Let ${\omega _n} = \cos \left( {\frac{{2\pi }}{n}} \right) + i\sin \left( {\frac{{2\pi }}{n}} \right)$ and ${i^2} = -1$. Then $(x + y{\omega _3} + z{\omega _3}^2)(x + y{\omega _3}^2 + z{\omega _3})$ is equal to:

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 $.....$ .

If $\omega$ is a cube root of unity,then the value of $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$ is:

If $x + \frac{1}{x} = 2\cos \alpha$,then $x^n + \frac{1}{x^n} = $