# How do you simplify (-1)^(1/3)?

Jan 7, 2016

It depends.

#### Explanation:

In Real arithmetic you don't have any choice:

${\left(- 1\right)}^{\frac{1}{3}} = - 1$

In Complex arithmetic you might prefer to define:

${\left(- 1\right)}^{\frac{1}{3}} = \cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2} i$

or you might prefer to stick with ${\left(- 1\right)}^{\frac{1}{3}} = - 1$.
It rather depends on the context.

$- 1$ has three cube roots:

$- 1$

$\frac{1}{2} + \frac{\sqrt{3}}{2} i$

$\frac{1}{2} - \frac{\sqrt{3}}{2} i$

If you are dealing with Complex arithmetic then you commonly want to be aware of all three.