Reduction of a complex number

Question

How can I reduce the expression i¹²⁷ to the lowest positive power of i?

Answer 1

i¹ = i, i² = -1, i³ = -i, i⁴ = 1, i⁵ = i⁴i¹ = i, i⁶ = i⁴i² = -1, i⁷ = i⁴i³ = -i, i⁸ = i⁴i⁴ = 1, ...

The powers cycle through i, -1, -i, 1. To simplify i¹²⁷, divide 127 by 4 to get a quotient of 31 with remainder 3.

i¹²⁷ = i^{4(31) + 3} = (i⁴)²⁷i³ = i³ = -i

Answer 2

i^127

= i^3(i^124) = i^3(1) = i^3

i to any power divisible by 4= 1