If $i=\sqrt{-1}$,then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

Let $p(x) = a_0 + a_1 x + \ldots + a_n x^n$ be a non-zero polynomial with integer coefficients. If $p(\sqrt{2} + \sqrt{3} + \sqrt{6}) = 0$,then the smallest possible value of $n$ is

If ${a^2} + {b^2} = 1,$ then $\frac{{1 + b + ia}}{{1 + b - ia}} = $

Let $\alpha, \beta$ be the roots of the equation $x^2-x+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then $\alpha^6+\alpha^4+\beta^4-5 \alpha^2$ is equal to

The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is

The number of solutions for $z^3+\bar{z}=0$ is