Question

how (-1)^(1/3) = 0.5+0.866i
http://www.wolframalpha.com/input/?i=%28-1%29%5E%281%2F3%29

Answer 1

Roots don't work very well for negative numbers. In general, if you take some root of a negative number, you get a complex number. Because of that, Wolfram Alpha always uses that form.

Answer 2

i wasn't sure about cube root. X^3 = -1 should have three roots right? one of them is -1, and the rest are?

Answer 3

It has a maximum of 3 distinct roots. This has a list of all three roots. http://www.wolframalpha.com/input/?i=x%5E3%2B1

Answer 4

oh ... i can see

Answer 5

If you write a complex number as
\[ Ae^{i\theta} \]
then for example the cube root is
\[ A^{\frac{1}{3}}e^{i\frac{\theta}{3}} \]
But because \( e^{i2\pi} = 1\)
we could rewrite
\[Ae^{i\theta}= Ae^{i2\pi}e^{i\theta} =Ae^{i(\theta+2\pi)} \]
and the cube root is
\[ A^{\frac{1}{3}}e^{i\frac{\theta+2\pi}{3}}\]
This is the second of the 3 roots.
We can add another 2pi to theta to get the 3rd root.

But notice if we add 6pi to theta:
\[ \frac{\theta+6\pi}{3}=\frac{\theta}{3} +2\pi \to\frac{\theta}{3} \]
and we have come full circle. There will only be 3 distinct roots.