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# 4.10: Powers and Roots of Complex Numbers

Difficulty Level: At Grade Created by: CK-12
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Practice Powers and Roots of Complex Numbers

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Estimated17 minsto complete
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Estimated17 minsto complete
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### Vocabulary Language: English

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.

De Moivre's Theorem

De Moivre's theorem is the only practical manual method for identifying the powers or roots of complex numbers. The theorem states that if $z= r(\cos \theta + i \sin \theta)$ is a complex number in $r cis \theta$ form and $n$ is a positive integer, then $z^n=r^n (\cos (n\theta ) + i\sin (n\theta ))$.

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