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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