nth roots of a complex number

Unlike the reals, where the $n$th root may not exist or may be unique, in the complex numbers every non-zero number always has exactly $n$ distinct $n$th roots.

Property — $n$th roots of a complex number

Given $z=\rho\,e^{i\theta}$ with $\rho>0$, the $n$th roots of $z$ are exactly $n$, given by: $w_k = \sqrt[n]{\rho}\,\exp\!\left(i\,\frac{\theta + 2k\pi}{n}\right), \qquad k = 0, 1, \ldots, n-1.$ These $n$ roots are the vertices of a regular polygon with $n$ sides inscribed in the circle centred at the origin with radius $\sqrt[n]{\rho}$.

All the roots have the same modulus $\sqrt[n]{\rho}$ and arguments differing by $2\pi/n$: for this reason they are arranged at the vertices of a regular polygon.

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Topics: Complex numbers
Concepts: Exponential form · nth roots
Methods: Numeri complessi polare
Skills: Use formulae