Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 768: 55


The power of the complex numbers in the rectangular form is $-4-4\sqrt{3}i$.

Work Step by Step

Here, $z={{\left[ 2\left( \cos {{80}^{{}^\circ }}+i\sin {{80}^{{}^\circ }} \right) \right]}^{3}}$ Therefore, $\begin{align} & z={{\left[ 2\left( \cos {{80}^{{}^\circ }}+i\sin {{80}^{{}^\circ }} \right) \right]}^{3}} \\ & z={{2}^{3}}\left( \cos 3\times {{80}^{{}^\circ }}+i\sin 3\times {{80}^{{}^\circ }} \right) \\ & z=8\left( \cos {{240}^{{}^\circ }}+i\sin {{240}^{{}^\circ }} \right) \\ & z=8\left( -\frac{1}{2}-i\frac{\sqrt{3}}{2} \right) \\ \end{align}$ Simplify it further, to get, $z=-4-4\sqrt{3}i$ The power of the complex number in the rectangular form is $z=-4-4\sqrt{3}i$
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