solve a math problem?

Question

whats i^1066
(the ^ sign means exponent, 1066 is the exponent. i to the 1066)
(please show your steps)

Answer 1

Best way to solve this: visualize a graph, where the x-axis is the real number line and the y-axis is the imaginary line. So, (1,0) is still (1,0), but (0,1) is (0, i). If you add 1 to the exponent of i, you get i^2, which is a 90 degree rotation from (0,i) to (-1,0). Hope I"m making sense here. So, you start yourself at basically (0,i), and make 1066 rotations. BTW i^5 = i, where u started, because you just made a complete circle to i again.
Anyways, to find the answer:
1066/4 = 266 R 2, which means you make 266 circles back to i, but you have 2 extra 90 degree trips to make.
So, you start at (0,i) and move 180 degrees across to (0,-i).
-i is your answer.

Answer 2

1

Answer 3

Unless I am mistaken, i is the square root of -1. So any odd exponent would yield -1, while an even one would yield 1. So...1.

EDIT - Upon seeing JZ's response, I realized that I had indeed made an error. (Don't be too hard on me, I was thinking quickly, and thought of -1, rather than the square root of -1.) But it's still not nearly as complicated as he suggests, and I do not believe he arrived at the correct answer.

i = sqrt -1, or -1^(1/2)
so i^0=1
i^1=i
i^2=-1
i^3=-i
i^4=1
i^5 = i^2 x i^3 = i
i^6 = (i^2)^3 = -1

As you can see, this pattern continues. Even numbers that are multiples of 4 are equal to 1, even numbers that are not multiples of 4 (such as 1066) are equal to -1.

i^1066 = (i^2)^533 = -1^533 = -1

The answer is -1.

Answer 4

thats hard i dont know