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.

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, …

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