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Geometrically, we are multiplying the lengths and add their angles. \\  We need to find values of $z$ where it is multiplied by itself $n$ times and is equal to $re^{i\theta}.$ \\So, let $z=se^{i\theta}.$\\ Thus, we have $s^n=r.$\\ Hence, $s=\sqrt[n]{r}$\\  To solve for $\theta$, we have $n\theta= \theta +2\pi k$, for some $k\in\mathbb{Z}$ \\  Then $\theta=\frac{\theta + 2 \pi k}{n}$ k}{n}$\\  Therefore we have $z=\sqrt[n]{r}e^{ie^{\frac{\theta + 2 \pi k}{n}}  \end{proof}  \end{problem}