Question

How do I find the fifth root of a complex number?

Answer 1

Convert to polar form first, then...

Explanation:

A Complex number in the form `r(cos theta + i sin theta)` has `5`th roots:

`root(5)(r)(cos (theta/5) + i sin (theta/5))`

`root(5)(r)(cos ((theta + 2pi)/5) + i sin ((theta+2pi)/5))`

`root(5)(r)(cos ((theta + 4pi)/5) + i sin ((theta+4pi)/5))`

`root(5)(r)(cos ((theta + 6pi)/5) + i sin ((theta+6pi)/5))`

`root(5)(r)(cos ((theta + 8pi)/5) + i sin ((theta+8pi)/5))`

Conventionally your original `theta` is in the range `(-pi, pi]` or the range `[0, 2pi)` according to your definition of `Arg(z)` and the first of these five roots is the Principal Complex fifth root.