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I'm an Android programmer and am working on a graphing calculator. I have been looking for the limits on which roots can be done. I have a decent understanding of mathematics but can not seem to find these limits. Any help would be great, thanks.

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    Sorry, missed the negative signs my mistake.2012-03-02

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Let $x < 0$ and $y \in \mathbb{R}$. Note that $\text{Arg } x = \pi$.

By the definition of the complex logarithm we have $x^{1/y} = e^{(\log x)/y} = e^{[\log|x| + i(\pi+2\pi\ell)]/y} = e^{(\log |x|)/y} e^{i\pi(1+2\ell)/y}$ where $\ell$ is any integer. Thus $x^{1/y}$ has a real value if and only if $(1+2\ell)/y$ is an integer for some $\ell$. This happens exactly when $y = (1+2m)/n$ for some integers $m$ and $n$ with $n \neq 0$.

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    @J.d.: Ah, yeah, definitely.2012-03-02
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Given a real number $x < 0$ and an integer $y \in \mathbb{Z}$. The $y$th root of $x$, given by: $ r = x^{\frac{1}{y}} $ is a real number if and only if $y$ is an odd number.

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    In my above comment I didn't mean to use $k$ twice... instead take $y = (1+2n)/m$.2012-03-02