Taking a complex root divides the angle in several different ways.
WHAT HAPPENS IF…How many distinct cube roots does i have?
Interactive exploration
Loading the mathematical model…
Connect the picture to the ideazk=r1/nei(θ+2kπ)/n,k=0,…,n−1
Writing a nonzero target as re^{iθ}, its nth roots must have magnitude r^{1/n}. Their nth powers must have arguments θ modulo 2π, giving n equally spaced choices. Raising each displayed vector to the nth power returns the same target.
A detail that matters
A nonzero complex number has n distinct nth roots; the zero target has only the single root zero.