# Because roots of unity is too mainstream

Algebra Level 5

Consider the set of all complex $$n^\text{th}$$ roots of $$x$$, $$\displaystyle \{a_i\}_{i=1}^{i=n}$$ where $$x\in\Bbb{C}$$.

We further denote the summation $$\displaystyle P_m = \sum_{j=1}^n \left(a_j\right)^m$$.

Find an expression for $$P_{nk}$$.

Note: $$m,n,k$$ are all positive integers.

Bonus: Can you write a result which provides all values of $$P_m$$?

Clarification: $$\Bbb C$$ is the set of all complex values.

Inspired by this problem.

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