$(1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}})^5=$

The two numbers such that each one is the square of the other are:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 27 = 0$,find the quadratic equation whose roots are $\left( \frac{\gamma}{\alpha} \right)^2$ and $\left( \frac{\beta}{\alpha} \right)^2$.

If $\omega$ represents a complex cube root of unity,then $\left(1+\frac{1}{\omega}\right)\left(1+\frac{1}{\omega^2}\right)+\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^2}\right)+\ldots+\left(n+\frac{1}{\omega}\right)\left(n+\frac{1}{\omega^2}\right)=$

If $n$ is a positive integer not a multiple of $3$,then $1 + \omega^n + \omega^{2n} = $

If $z^{2} + z + 1 = 0$,$z \in \mathbb{C}$,then $\left| \sum_{n=1}^{15} \left( z^{n} + (-1)^{n} \frac{1}{z^{n}} \right)^{2} \right|$ is equal to