Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 1 - Equations and Inequalities - 1.3 Complex Numbers - 1.3 Exercises - Page 113: 103

Answer

$(\displaystyle \frac{\sqrt{3}}{2}+\frac{1}{2}i)^{3}=i$ So,$\displaystyle \frac{\sqrt{3}}{2}+\frac{1}{2}i$ is a cube root of $i$.

Work Step by Step

$(\displaystyle \frac{\sqrt{3}}{2}+\frac{1}{2}i)^{3}\qquad$...Combine the fractions $(\displaystyle \frac{\sqrt{3}+i}{2})^{3}\qquad$...Apply exponent rule: $(\displaystyle \frac{a}{b})^{n}=\frac{a^{n}}{b^{n}}$ $=\displaystyle \frac{(\sqrt{3}+i)^{3}}{2^{3}}\qquad$...Apply Perfect Cube Formula: $(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3},\ a=\sqrt{3},\ b=i$. $=\displaystyle \frac{(\sqrt{3})^{3}+3(\sqrt{3})^{2}i+3\sqrt{3}i^{2}+i^{3}}{2^{3}}\qquad$...Simplify. $=\displaystyle \frac{3\sqrt{3}+9i-3\sqrt{3}+i^{3}}{8}\qquad$...Group like terms $=\displaystyle \frac{i^{3}+9i+3\sqrt{3}-3\sqrt{3}}{8}\qquad$...Simplify. $=\displaystyle \frac{i^{3}+9i}{8}\qquad$...Simplify. $i^{3}=i^{2}\cdot i=-1\cdot i=-i$ $=\displaystyle \frac{-i+9i}{8}$ $=\displaystyle \frac{8i}{8}$ $=i$ Therefore, $(\displaystyle \frac{\sqrt{3}}{2}+\frac{1}{2}i)^{3}=i$
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