Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 7 - Section 7.7 - Complex Numbers - 7.7 Exercises - Page 492: 88



Work Step by Step

In order to calculate the larger powers of $i$, we can first calculate the values of $i$, $i^{2}$, $i^{3}$, and $i^{4}$. $i^{1}=i$ $i^{2}=i\times i=-1$ $i^{3}=i\times i^{2}=i\times-1=-i$ $i^{4}=i^{2}\times i^{2}=-1\times-1=1$ We can use these values and the rules of exponents to derive larger powers of $i$. Therefore, $i^{-17}=\frac{1}{i^{17}}=\frac{1}{i^{16}\times i^{1}}=\frac{1}{(i^{4})^{4}\times i^{1}}=\frac{1}{1^{4}\times i}=\frac{1}{1\times i}=\frac{1}{i}=\frac{1(-i)}{i(-i)}=\frac{-i}{-i^{2}}=\frac{-i}{-(-1)}=\frac{-i}{1}=-i$.
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