Finding all the cube (or other) roots of a number

Finding all the cube (or other) roots of a number N:

Remember that in ℂ every number has three cube roots, four 4th roots, five 5th roots, 57 57th roots…. and on the Argand diagram (graph) they look like evenly-spaced spokes of a wheel, with one spoke along the x-axis.


N = N cis 0 = N cis 2π = N cis 4π = N cis 6&6pi; +….

(as many different ways of writing N as you have roots)

If n is the positive real k’th root of N (example: 2 is the real fifth root of 32, 3 is the positive real fourth root of 81), then the roots are

n cis 0 (which is just n)

n cis 2π/k

n cis 4π/k

n cis 6π/k

and so on until you have k of them.

Draw the roots to check your working.


Example: the cube roots of 1 are 1, cis 2π/3, and cis 4π/3 (which is the same as cis −2π/3 )

If you want the roots in x+iy form, convert them using Rec( , ) on your calculator.