Simplify the expression. Type the answer in the standard form of complex numbers. i^15

Answer 1

i = √-1

i¹ = i

i² = -1

i³ = i² · i = -i

i⁴ = i² · i² = (-1)(-1) = 1

i⁵ = i⁴ · i = (1)(i) = i

i⁶ = i⁴ · i² = (1)(-1) = -1

i⁷ = i⁴ · i³ = (1)(-i) = -i

i⁸ = i⁴ · i⁴ = (1)(1) = 1

Notice the pattern. Always goes i, -1, -i, 1, i, -1, -i, 1, ...

All of the i⁴ = 1, so any multiple of 4 i's = 1 and so only the remainder (n/4) indicates the answer.

power/4 = ____ remainder = 1, answer is i

power/4 = ____ remainder = 2, answer is -1

power/4 = ____ remainder = 3, answer is -i

power/4 = ____ remainder = 0, answer is 1

continuing

i⁹ = i⁴ · i⁴ · i = (1)(1)(i) = i OR 9/4 = 2 rem=1, rem=1 means answer = i

i¹⁰ = i⁴ i⁴ ·· i² = (1)(1)(-1) = -1 OR 10/4 = 2 rem = 2, rem=2 means answer is -1

i^·11 = i⁴ i⁴ · i³ = (1)(1)(-i) = -i OR 11/4 = 2 rem = 3, rem=3 means answer is -i

i¹²· = i⁴ · i⁴ i⁴ = (1)(1)(1) = 1 OR 12/4 = 3 rem = 0, rem=0 means answer is 1

15/4 = 3, rem=3, so i¹⁵ = -i