## Algebra and Trigonometry 10th Edition

Published by Cengage Learning

# Chapter 8 - 8.6 - Trigonometric Form of a Complex Number - 8.6 Exercises - Page 613: 54

#### Answer

$[3(cos~60°+i~sin~60°)]^4=-\frac{81}{2}-\frac{81\sqrt 3}{2}i$

#### Work Step by Step

DeMoivre's Theorem: If $z=r(cos θ+i~sin θ)$, then $z^n=r^n(cos~nθ+i~sin~nθ)$ $z=3(cos~60°+i~sin~60°)$ $z^4=3^4[cos~(4 ·60°)+i~sin~(4 ·60°)]$ $z^4=81(cos~240°+i~sin~240°)$ $z^4=81[-\frac{1}{2}+i(-\frac{\sqrt 3}{2})]$ $z^4=-\frac{81}{2}-\frac{81\sqrt 3}{2}i$

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