Answer: Choice A) i
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Step-by-step explanation:
Let's look at the pattern of powers of i
- i^0 = 1
- i^1 = i
- i^2 = (sqrt(-1))^2 -1
- i^3 = i*i^2 = i*(-1) = -i
- i^4 = (i^2)^2 = (-1)^2 = 1
We loop back to 1, and the process starts all over again. The cycle is 4 units long. Ie, the pattern repeats every 4 terms.
What this means is that we can use remainders to quickly figure out something like i^41 without having to use a calculator or tons of paper.
Dividing 41/4 leads to a quotient of 10 and a remainder of 1. We only care about the remainder. The remainder 1 means i^41 = i^1 = i
Note how 41 and 1 lead to the same remainder when we divide by 4.