Non-integer exponents of negative numbers?

Question

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 1

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 $m$.

Answer 2

If $n$ is not an integer, then $(-1)^n$ is not a real number.

Solving this problem requires the identity: $$ e^{i\theta} = \cos \theta + i \sin \theta $$

Thus, $$-1 = \cos \pi = e^{i\pi}$$

So $$ (-1)^n = (e^{i\pi})^n = e^{i n \pi} = \cos n\pi + i \sin n \pi$$

And, in general, for $m < 0$: $$ m^n = |m| \cos n\pi + i |m| \sin n\pi $$

This works for any real $n$, not just integers.