write each given power of i in simplest terms?

Question

1. i^12
2. i^7
3. i^ 49
4. i^72
5. i^54
6. i^99
7. i^300
8. i^246
9. i^91
10. i^2001

Answer 1

First divide your exponent by four and find your remainder.
Your remainder will become the exponent, except when it is 0.

Remainder of 1: i^1= √(-1)
Remainder of 2: i^2= -1
Remainder of 3: i^3= -i
Remainder of 0: i^4= 1

Answer 2

I'm not gonna do all of them but I'll do the first two. Any i to the multiple of 4 =1. So the first one is 1 since 4*3=12. The second one try to make it into multiples of 4 so i^4*i^3=1*-i. So it's -i. All you really need to know is any i to a multiple of 4=1 and i^2=-1 and i^3 is =-i and you can go from there.

Answer 3

The powers of i repeat every 4 numbers.
We have i^0 =1
i ^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
and the cycle starts again.
So let's look at i^n:
If n = 4k, then i^4k = (i^4)^k = 1
if n = 4k+1, then i^4k+1 = i^4k*i = i
if n = 4k+2, then i^4k+2 = i^4k*i^2 = i^2 = -1
finally
if n = 4k+3, then i^4k+3 = i^4k*i^3 = i^3 = -i.
So we can go right down your list and read off the answers:
1 1
2 -i
3 i
4 1
5 -1
6 -i
7 1
8 -1
9 -i
10 i

Answer 4

i^4n = 1
i^(4n + 1) = i
i^(4n + 2) = -1
i^(4n + 3) = -i