College Algebra & Trigonometry, 2018a

1.4. COMPLEX NUMBERS 43
Powers of i
The powers of i follow an interesting pattern based on the definition that i 2 = −1.
We can see that i 1 = i and that i 2 = −1, as a result, i 3 = i 2 ∗ i 1 = −1 ∗ i = −i.
In a similar fashion, i 4 = i 2 ∗ i 2 =(−1)(−1) = 1.
This means that i 5 = i 4 ∗ i =1∗ i = i.
If we put all of this information together we get the following:
i 1 = i
i 2 = −1
i 3 = −i
i 4 =1
i 5 = i 1 = i
i 6 = i 2 = −1
i 7 = i 3 = −i
i 8 = i 4 =1
In other words, every power of i is equivalent to either i, −1, −i, or 1. To determine
which of these values a power of i is equivalent to, we need to find the
remainder of the exponent when it is divided by 4.
Example
Simplify i 38
Since every i 4 =1, then i 38 = i 36 ∗ i 2 =(i 4 ) 9 ∗ i 2 =1 9 ∗ i 2 = i 2 = −1
Since 38 is 2 more than a multiple of 4, then i 38 = i 2 = −1.