Powers of i
Now we will examine an interesting property of i. When we raise it to any
positive integer power and simplify, the result is one of only four
possibilities: i, -1, -i, or 1.
Look at the powers of i listed in the table.
To simplify a higher power of i, we use this fact: i4 = 1.
| For example, let’s simplify i10.
|
i10 |
| Use the Multiplication Property of Exponents
to write i10 as a product where one factor is a
power of i that is a multiple of 4.
|
= i8 · i2 |
| Rewrite i8 in terms of i4.
Replace i4 with 1. Replace i2 with -1.
Multiply.
So, i10 = -1. |
= (i4)2
· i2 = (1)2
· i2
= -1 |
Note:
i1 = i
i2 = -1
i3 = i2 · i = (-1)
· i = -i
i4 = i2 · i2
= (-1)(-1) = 1
i5 = i4 · i1
= 1 · i = i
i6 = i4 · i2
= 1 · (-1) = -1
i7 = i4 · i3
= 1 · -i = -i
i8 = i4 · i4
= 1 · 1 = 1
i9 = i4 · i4
· i = 1 · 1
· i = i
The pattern repeats:
i, -1, -i, 1, i, -1, -i, 1, …
We can follow the same process to simplify i27.
| Write i27 using a multiple of 4. Rewrite i24 in terms of i4.
Replace i4 with 1.
Replace i3 with -i.
Multiply. |
i27 |
= i24
· i3 = (i4)6
· i3
= 16 · i3
= 1 · (-i)
= -i |
So, i27 = -i.
Example 1
Simplify.
a. i35
b. i82
c. i20
Solution
| a. To simplify i35, divide 35 by 4.
The result is 8 with remainder 3. |
 |
|
i35 |
= (i4)8
· i3
= 18 · i3
= 1 · (-i)
= -i |
| b. To simplify i82, divide 82 by 4. The result is 20 with remainder 2. |
 |
|
i82 |
= (i4)20
· i2
= 120 · i2
= 1 · (-1)
= -1 |
| c. To simplify i20, divide 20 by 4.
The result is 5 with remainder 0. |
 |
|
i20 |
= (i4)5
· i0
= 15 · 1
= 1 · 1
= 1 |
|
