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Powers of Complex Numbers
De Moivre's Theorem for powers and roots of complex numbers.
De Moivre's Theorem for powers and roots of complex numbers.
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Now that we know how to represent complex numbers in the form ​reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
And since ​reiθ=rcisθ:
De Moivre's Theorem can also be used to find the ​nth​ roots of complex numbers:
or equivalently
However, since ​cisθ=cis(θ+2kπ)​ for any ​k∈Z, then we actually have
Note that ​k​ stops at ​n−1​ since when ​k=n​ we have