---
tag: 0YM6P6C206
kind: statement
role: theorem
name: Existence of nth roots of complex numbers
msc: [12D99]
uses: [0BRXFNQ5JV, 0E021GF2B8]
sources:
  - url: https://en.wikipedia.org/wiki/Nth_root
    locator: Complex roots
  - url: https://en.wikipedia.org/wiki/Complex_number
    locator: Powers and roots
---

## Statement

Let $z$ be a complex number and let $n$ be a positive integer.
Then there is a complex number $w$ with $w^n = z$.

## Remarks

For $z = r e^{i \theta}$ with $r > 0$, one solution is $w = r^{1/n} e^{i \theta / n}$, by the [[polar form|0BRXFNQ5JV]].
For $z \neq 0$, there are exactly $n$ solutions. They are $w$ times the $n$th [[roots of unity|0E021GF2B8]].
This fact is a special case of the fundamental theorem of algebra.

## Explanations

### L1 human

Every complex number has a square root, a cube root, and so on.
For example, $-1$ has the square roots $i$ and $-i$, and $i$ has the square roots $\pm \frac{\sqrt{2}}{2}(1 + i)$.
To take an $n$th root, take the $n$th root of the length and divide the angle by $n$.