nth Root of Complex Number Calculator

Every nonzero complex number has exactly $n$ distinct $n$th roots. For example, the number 8 has three cube roots: $2$, $-1 + i\sqrt{3}$, and $-1 - i\sqrt{3}$. The fundamental theorem of algebra guarantees that $z^n = w$ always has $n$ solutions in the complex plane.

Each root has the same modulus $r^{1/n}$ but a different argument. Consecutive roots differ by an angle of $\frac{2\pi}{n}$ radians ($\frac{360^\circ}{n}$). This equal spacing means the $n$ roots form the vertices of a regular $n$-gon centered at the origin.

The $n$th roots of unity are the $n$ solutions to $z^n = 1$. They lie on the unit circle (radius 1) at angles $\frac{2\pi k}{n}$ for $k = 0, 1, \ldots, n - 1$. The primitive root $\omega = e^{2\pi i/n}$ generates all others as powers: $1, \omega, \omega^2, \ldots, \omega^{n-1}$.

Write the number in polar form first. For $-8$: $r = 8$ and $\theta = \pi$. The three cube roots have modulus $8^{1/3} = 2$ and arguments $\frac{\pi + 2\pi k}{3}$ for $k = 0, 1, 2$. This yields $2(\cos 60^\circ + i\sin 60^\circ) = 1 + i\sqrt{3}$, the real root $-2$, and $1 - i\sqrt{3}$.

Complex $n$th roots appear in signal processing (the discrete Fourier transform uses roots of unity), electrical engineering (AC circuit analysis), control theory (pole placement), and polynomial factoring (splitting $x^n - c$ into linear factors).