exponentiation - Non-integer exponents of negative numbers?

There is a formula for exponents of negative numbers as follows:

$m^n=(-1)^n|m|^n$.

This formulation works when $m<0$ and $n\in \mathbb{Z}$. But what about for $n\in \mathbb{R}$? Is there a simple way to define non-integer exponents of negative numbers?

Answer

In general, no. But negative numbers have well-defined cube roots, for instance. Specifically, if $n$ can be expressed as a rational number with an odd denominator, then $m^n$ is well-defined for all $m \in \mathbb R$.

Otherwise there is no consistent way to define $m^n$ for negative $m$.