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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