$(\sqrt{\sqrt{2}+1} + i\sqrt{\sqrt{2}-1})^8 =$

For $n > 1$ and $n \in N$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n = z^n$, then $\sum_{i=1}^{n-1} \frac{\cot^{-1}(2|\operatorname{Im} z_i|) - 1}{2 \operatorname{Re} z_i} = $

If $\alpha, \beta$ are the roots of the equation $x^2-4x+8=0$,then for any $n \in N$,$\alpha^{2n}+\beta^{2n}$ equals

If $\omega$ is an imaginary cube root of unity,then the value of $\sin \left[ (\omega^{10} + \omega^{23})\pi - \frac{\pi}{4} \right]$ is

If $\alpha, \beta, \gamma$ are the cube roots of $p$ $(p < 0)$,then for any $x, y$ and $z$,$\frac{x\alpha + y\beta + z\gamma}{x\beta + y\gamma + z\alpha} = $

If $z = \cos \alpha + i \sin \alpha$; $0 < \alpha < \frac{\pi}{4}$,then $\left|\frac{1+z^4}{1-z^3}\right| = $