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Veva Garcia edited untitled.tex
almost 8 years ago
Commit id: ce1bc212cf8c0503f90dd45f3031ab5c6e10c925
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\begin{proof}
Let $n \in \mathbb{Z}$ and $w \in \mathbb{C}.$ So, in polar coordinates, $w=re^{i\theta}$ for some $r,\theta \in \mathbb{R}.$ Notice that the $r$ represents the modulus of the complex number and $\theta$ is the argument. So, we have $z^n=re^{i\theta}.$\\
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, \\So, let
$z=se^{i\theta}.$ $$z=se^{i\theta}.$$ Thus, we have
$s^n=r.$ $$s^n=r.$$ Hence,
$s=\sqrt[n]{r}$ $$s=\sqrt[n]{r}$$
\end{proof}
\end{problem}