# Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.4 De Moivre's Theorem: Powers and Roots of Complex Numbers - 8.4 Exercises - Page 382: 12

$(\frac{-27}{2}+i.\frac{27\sqrt 3}{2})$

#### Work Step by Step

De Moivre’s Theorem states that when $r (\cos\theta+i \sin\theta)$ is a complex number, and if $n$ is any real number, then the following relationship holds. $[ r (\cos\theta+i \sin\theta)]^{n}=[ r^{n} (\cos n\theta+i \sin n\theta)]$ In compact form, this is written $[ r cis\theta]^{n}=[ r^{n} (cis \theta)]$ $[ 3 cis40^{\circ}]^{3}=[ 3^{3} (cis 40^{\circ})]$ $[27 (\cos 40^{\circ}+i \sin 40^{\circ})]$ $=[ 27(\frac{-1}{2}+i.\frac{\sqrt 3}{2})]$ $=(\frac{-27}{2}+i.\frac{27\sqrt 3}{2})$

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