De Moivre’s Theorem and Powers of Complex Numbers

The polar form of complex numbers enables you to find powers of complex numbers. To find the square of a complex number, you use the rules for multiplication that were discussed previously. For example:

(r cis α)2 = (r cis α)(r cis α) = r · r · cis+ α) = r2cis

To find the cube of a complex number, use the square of the number and the same rule:

(r cis α)3 = (r cis α)2(r cis α) = (r cis 2α)(r cis α) = r2 · r · cis(2α + α) = r3cis

You can generalize the previous pattern to obtain a general formula for powers of complex numbers. This formula is called De Moivre’s Theorem.

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