If ${x_r} = \cos \left( \frac{\pi }{2^r} \right) + i\sin \left( \frac{\pi }{2^r} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots \infty$ is:

$(1-i \sqrt{3})^{2025}=$

${\left[ {\frac{{1 + \cos (\pi /8) + i\sin (\pi /8)}}{{1 + \cos (\pi /8) - i\sin (\pi /8)}}} \right]^8}$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$,then $\alpha^{2009}+\beta^{2009}$ is equal to

If $x^6 = (\sqrt{3} - i)^5$,then the product of all of its roots is

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$ such that $\alpha+\beta=-1, \gamma+\delta=1, \alpha^2=\beta$ and $\gamma^2=-\delta$,then $\alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022}=$