Finding nth Roots of a Complex Number
Simple Explanation
Every nonzero complex number has exactly n distinct nth roots. Written in trigonometric form r(cosθ+isinθ), its n roots all share the same modulus r^(1/n), but their angles are (θ+360°k)/n for k=0,1,...,n−1 — evenly spaced 360°/n apart around a circle.
Why Do We Need It?
This is a genuine surprise compared to real numbers (where, say, 8 has only ONE real cube root) — in the complex numbers, every nonzero number has exactly n nth roots, no more and no fewer.
See It
Three points evenly spaced 120 degrees apart on a circle of radius 2, representing the three cube roots of the complex number 8
Formula
The nth Roots of a Complex Number
zₖ = r^(1/n) [cos((θ+360°k)/n) + i sin((θ+360°k)/n)], for k = 0, 1, ..., n−1
Every nonzero complex number has exactly n distinct nth roots, all with the same modulus r^(1/n), evenly spaced 360°/n apart around a circle.
- k
- — an integer from 0 to n−1, giving one distinct root each
- r^(1/n)
- — the (positive, real) nth root of the modulus — the same for every root
When to use it: Whenever all n solutions to zⁿ = (some complex number) are needed.
Worked Example
Find all cube roots of a complex number
Find all cube roots of 8 (that is, solve z³=8).
Why Does This Work?
Raising any of these three roots to the 3rd power, by De Moivre's Theorem, gives modulus 2³=8 (correct) and angle 3×(0°,120°, or 240°) = 0°, 360°, or 720° — all of which are the SAME direction as 0° (since angles that differ by a full 360° describe the same point), confirming every one of these three roots genuinely cubes back to 8.
Real-Life Example
Finding all stable states in a rotational system
An engineer analyzing a system with rotational symmetry (like a signal that repeats every 1/n of a cycle) needs every equally-valid solution, not just one.
Finding all n complex roots directly captures every one of these physically distinct but equally valid states, evenly distributed around a cycle.
Practice
Find the modulus of each of the four 4th roots of 16.
HardCommon mistake
Finding only ONE root and stopping — every nonzero complex number has exactly n distinct nth roots, evenly spaced 360°/n apart, and all of them must be listed for a complete answer.
Quick Review
- Every nonzero complex number has exactly n distinct nth roots.
- All n roots share the same modulus, r^(1/n).
- The roots' angles are (θ+360°k)/n for k=0,1,...,n−1 — evenly spaced around a circle.