I know that if $m$ is an integer, then the $n$th root of $m$ must be an integer, else it is irrational. But what if $n$ and $m$ both are decimals - is there a way of easily telling if the $n$th root of $m$ is rational or irrational without actually calculating all/a lot the digits?
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3If $a/b$ is an irreducible fraction, then $\sqrt[m]{a/b}$ is rational iff $\sqrt[m]{a},\sqrt[m]{b}$ are integers. – Wojowu Mar 13 '16 at 20:41
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See the related question here. Yours is a bit more general though as it sounds like you are asking for noninteger roots as well, e.g. $e^{\frac{1}{\pi}}=\sqrt[\pi]{e}$. – JMoravitz Mar 13 '16 at 20:44