i² equals −1. That one identity is how powers of i cycle: i, −1, −i, 1. Complex numbers as a set are in complex numbers.
Reduce the power modulo 4
i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then repeat. i⁷ is the same as i³. A 2×2 determinant is a different leftover formula, in matrix determinant.
The short list is in ACT Math formulas. Exponent rules for real numbers are in exponent rules.
The bottom line
i² = −1. Higher powers: divide the exponent by 4 and read the remainder.
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