Final answer:
The powers of the imaginary unit i fall into a cycle: i to the first power is i, i squared is -1, i cubed is -i, and i to the fourth power is 1. Powers of i higher than 4 can be simplified by finding the remainder when the power is divided by 4 and then using the cycle of the first four powers to determine the equivalent power.
Step-by-step explanation:
The statement 'Every power of i that is greater than 4, is equivalent to either i, i², i³, or i⁴' refers to the properties of the imaginary unit i, which is defined as the square root of -1. The powers of i follow a cyclical pattern due to the nature of complex numbers in mathematics. Specifically, when calculating powers of i, we can use the following facts:
- i¹ = i
- i² = -1
- i³ = -i
- i⁴ = 1
Because i⁴ equals 1, when we calculate powers of i that are greater than 4, we end up in the same cycle: i⁵ is equivalent to i, i⁶ to i², and so on. That means any integer power of i can be reduced by finding the remainder when that power is divided by 4 (since the cycle repeats every 4 powers) and then using the above equivalences.