Simplify i^42?

Question

Please tell how you got the answer too because I need to understand it for a test coming up. Thanks so much.

Answer 1

i^42 = -1

Method:

i^42 = i^40 * i² = (i^4)^10 * (-1) = 1^10 * (-1) = -1

Answer 2

If you mean i=sqrt(-1), and want to consider what i^n is, let's start with lower powers and extrapolate from there.

n=0: i^0 = 1
n=1: i^1 = i
n=2: i^2 = [sqrt(-1)]^2 = -1
n=3: i^3 = i^2 * i = -1 * i = -i
----
n=4: i^4 = i^2 * i^2 = -1 * -1 = 1
n=5: i^5 = i^4 * i = 1 * i = i
n=6: i^6 = i^4 * i^2 = i * -1 = -i

So notice how n=4 gives the same answer as n=0 and how n=5 gives the same answer as n=1, etc. This means that the sequence is cyclical going {1, i, -1, -i, repeat}.

For higher values of n, you just need to figure out where in that cycle of four you fall. If n is divisible by four, then it's as if n=(0 or 4), so the answer is 1.

If n isn't divisible by 4, consider the remainder. If it's 1, then it's like n=(1 or 5) so the answer is i.

When the remainder is 2, it's like n=2, so the answer is -1. Finally when the remainder is 3, the answer is -i.

In your case: n = 42, which is not divisible by four. Instead, your remainder from 40 is two, which means your on the second point of the cycle. Hence, i^42 = -1.

Answer 3

Well i^4 = 1
42/4 = 10 with a remainder of 2 ( means that the cycle will repeat 10 times).
That means there are 2 "i"'s left
i*i = -1

So i^42 = -1

Answer 4

Sinnce i^4 = 1, we have
i^42 = (i^4)^10 * i² = i² = -1.

Answer 5

i one, i one, negatives in the middle.

1 i
2 -1
3 -i
4 1

So, divide 42 by 4, giving 10.5, the .5 corrolates to a remainder of 2, so the answer is -1.