The cube roots of 8

Example — The cube roots of 8

Since $8 = 8\cdot e^{i\cdot 0}$, applying the formula for the $n$th roots with $n=3$ and $\sqrt[3]{8}=2$ one obtains: $w_0 = 2\,e^{i\cdot 0} = 2,\quad w_1 = 2\,e^{i\cdot 2\pi/3} = 2\Bigl(-\tfrac12 + i\tfrac{\sqrt 3}{2}\Bigr) = -1 + i\sqrt{3},\quad w_2 = 2\,e^{i\cdot 4\pi/3} = -1 - i\sqrt{3}.$ The three roots form an equilateral triangle centred at the origin.

The three cube roots of 8 are the vertices of an equilateral triangle inscribed in the circle of radius $\sqrt[3]{8}=2$.

In general, the $n$ $n$th roots of a complex number are the vertices of a regular polygon with $n$ sides, inscribed in the circle of radius $\sqrt[n]{\rho}$. In the example, $n=3$ gives an equilateral triangle.

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Topics: Complex numbers
Concepts: Exponential form · Argand-Gauss plane · nth roots
Skills: Calculate · Use formulae