Why is i² equal to −1 and how do I simplify powers of i like i^50?

The imaginary unit i is defined as the square root of −1, so by definition i² = −1. This was introduced precisely because no real number squares to a negative, yet we needed solutions to equations like x² + 1 = 0. The powers of i cycle in a pattern of four: i¹ = i, i² = −1, i³ = −i, and i⁴ = 1, then it repeats. To find any power, divide the exponent by 4 and look only at the remainder. For i^50, divide 50 by 4 to get remainder 2, so i^50 equals i² which is −1. For i^27, the remainder is 3, so it equals i³ = −i. This remainder shortcut works for any high power and saves a lot of time in exams.