If $i{z^4} + 1 = 0$,then $z$ can take the value

The smallest positive integral value of $n$ such that $\left[\frac{1+\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}}{1+\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}}\right]^{n} = 1$ is:

If $\omega$ is a complex cube root of unity,then $\sin \left\{\left(\omega^{10}+\omega^{23}\right) \pi-\frac{\pi}{4}\right\}=$

If $\omega ( \neq 1)$ is a cube root of unity and $(1 + \omega )^7 = A + B\omega$,then $A$ and $B$ are respectively,the numbers

If $a=\cos \left(\frac{8 \pi}{11}\right)+i \sin \left(\frac{8 \pi}{11}\right)$,then $\operatorname{Re}\left(a+a^2+a^3+a^4+a^5\right)=$

One of the roots of the equation $x^{14}+x^9-x^5-1=0$ is