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 Depdendent Variable

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 Dependent Variable

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# Powers of Complex Numbers

To raise a complex number to a power, consider repeated use of the multiplication rule.

 z = r(cos θ + i sin θ) z2 = r2(cos 2θ + i sin 2θ) z3 = r3(cos 3θ + i sin 3θ)

This pattern leads to the following important theorem, which is named after the French mathematician Abraham DeMoivre (1667â€“1754).

Theorem

DeMoivreâ€™s Theorem

If z = r(cos θ + i sin θ) is a complex number and n is a positive integer, then

zn = [r(cos θ + i sin θ)]n =  rn(cos nθ + i sin nθ)

Example 1

Finding Powers of a Complex Number

Use DeMoivreâ€™s Theorem to find .

Solution

First convert to polar form.

Then, by DeMoivreâ€™s Theorem, you have

NOTE

Notice in Example 1 that the answer is a real number.