1. Aim: How can we simplify powers of i?
The powers of i repeat in a definite cycle: i, -1, -i, 1
By definition i0 = 1. Therefore:
i0 = i1 = i i2 = i3 = -i
i4 = i5 = i i6 = i7 = -i
i8 = i9 = i i10 = i11 = -i
i12 = i13 = i i14 = i15 = -i
Using
and i2 = -1.

2. Aim: How can we simplify powers of i?
If a whole number exponent is divided by 4, the remainder is 0, 1, 2, or 3.
We can simplify powers of i by using the remainders after dividing by 4.
Example:
Write i82 in simplest terms
82 = 20 remainder 2 4
Therefore i82 is equivalent to i2. So i82 = -1

3. Aim: How can we simplify powers of i?
Write each given power of I in simplest terms, as 1, i, -1, or -i
i12
i91
i49
i54

4. Aim: How can we simplify powers of i?
Write each given power of I in simplest terms, as 1, i, -1, or -i
i = i0 = 1
2. i = i3 = -i
3. i = i1 = i
4. i = i2 = -1

5. Go to the Practice!