Question

Explain, in complete sentences, the pattern of exponents on the imaginary number i. What is the usefulness of knowing this pattern when performing operations on complex numbers?

Answer 1

the patern of positive integer exponents repeats every 4
i^1=i; i^2=-1; i^3=-i; i^4=1
i^5=i; i^6=-1; i^7=-i; i^8=1
.
.
.

Answer 2

You can use this pattern to simplify i to any integr power by dividing by 4 to determine how many time you have been through the cycle and where you are in the next.

Examples:
i^25=i^1=1 (4 divides 25 6 times with a remainder of 1)
i^27=i^3=-i (4 divides 27 6 times with a remainder of 3)
etc.