How do I find all the nth roots of a complex number and why are there exactly n of them?

In one line

Write the number as \(r[\cos(\theta+2k\pi)+i\sin(\theta+2k\pi)]\) and apply De Moivre's theorem with power \(1/n\); taking \(k=0,1,\dots,n-1\) gives the \(n\) distinct roots.

Explained simply

A point on a circle can be reached by turning \(\theta\), or \(\theta+360^\circ\), or \(\theta+720^\circ\). These are the same point. But when you divide the angle by \(n\) to take a root, these different turns give different answers. After \(n\) of them the answers start repeating. That is why a number has exactly \(n\) different \(n\)th roots.

Technically, \(z^{1/n}=r^{1/n}\left[\cos\dfrac{\theta+2k\pi}{n}+i\sin\dfrac{\theta+2k\pi}{n}\right]\), \(k=0,1,\dots,n-1\). All roots have the same modulus \(r^{1/n}\) and are spaced \(2\pi/n\) apart, so they form a regular \(n\)-sided polygon centred at the origin.

How it works

1. Write \(z\) in polar form and add \(2k\pi\) to the argument.
2. Apply the power \(1/n\) (or \(p/q\)) using De Moivre's theorem.
3. Put \(k=0,1,\dots,n-1\) and write each root.
4. For \(z^{p/q}\) (in lowest terms), first find \(z^p\), then take \(q\) roots: \(r^{p/q}\left[\cos\dfrac{p\theta+2k\pi}{q}+i\sin\dfrac{p\theta+2k\pi}{q}\right]\), \(k=0,\dots,q-1\).
5. Continued product of all roots: multiply the exponential forms, i.e. add all the angles.

Key formula

\(z^{1/n}=r^{1/n}e^{i(\theta+2k\pi)/n},\ k=0,1,\dots,n-1\), where \(r^{1/n}\) is the positive real root of \(r\).

Quick example

Find the cube roots of \(8i\). Here \(r=8\), \(\theta=\pi/2\). Roots: \(2\left[\cos\dfrac{\pi/2+2k\pi}{3}+i\sin\dfrac{\pi/2+2k\pi}{3}\right]\).

• \(k=0\): angle \(\pi/6\): \(2\left(\frac{\sqrt3}2+\frac i2\right)=\sqrt3+i\)
• \(k=1\): angle \(5\pi/6\): \(-\sqrt3+i\)
• \(k=2\): angle \(3\pi/2\): \(-2i\)

Check: \((-2i)^3=-8i^3=8i\). Correct.

Common mistakes

• Writing only one root (\(k=0\)) and stopping.
• Taking \(k\) from 1 to \(n\) and then also \(k=0\), which repeats a root.
• For \(z^{p/q}\), dividing the angle by \(q\) before multiplying by \(p\) and losing roots.

How MU asks it

Common 5/6-mark parts: "Find all the values of \((1+i)^{1/3}\)", "Find the continued product of all values of \(\left(\frac12+\frac{\sqrt3}{2}i\right)^{3/4}\)" (answer: 1, because the four values have angles \(\pi/4, 3\pi/4, 5\pi/4, 7\pi/4\), which add to \(4\pi\), and \(\cos4\pi+i\sin4\pi=1\)), or "Find the cube roots of unity and show their sum is zero". Write the general root with \(2k\pi\), state the range of \(k\), list every root, and simplify.