Writing Complex Roots of Negative Numbers

[Arman777]

Let us suppose I have a number $x$ such that $x<0$. If I want to write the roots of the $x^{1/n}$. How can we write the roots of this number. I thought we can write

$$|x|^{1/n}e^{i\pi\theta}$$ for $\theta = \frac{2l + 1}{n}$ and $l = 0,1,2$ etc.

Is this correct ?

Similary If I wanted to write $x^{m/n}$, I should I write

$$|x|^{m/n}e^{mi\pi\theta}$$ for $\theta = \frac{2l + 1}{n}$ ?

 

[mathman]

I didn't check details, but your approach is correct.