Precise statement of exponent laws for real numbers Consider the "obvious" equality
$$(a^m)^n = a^{mn}.$$  In particular, if I only want to consider real numbers, for what values of $a,m,n$ is the above displayed equality true?  And where can I find the precise statement written down?  My first instinct was that as long as $a$, $a^m$, and $(a^m)^n$ were real numbers, then the equality would hold.  This would have ruled out examples like $((-1)^{1/2})^2$.  But we also have $$((-1)^2)^{1/2} \neq (-1)^1.$$  
I'm pretty sure the displayed equation is always true if we require $a$ to be positive, but it's also true for values like $a = -1$, $m = 1/3$, and $n = 3$. 
My guess is that the most general statement (again, only considering real numbers) requires $a$, $a^m$, $(a^m)^n$, and also $a^n$ to be real numbers.  Is this "easy" stuff written down in something like Rudin's Principles of Mathematical Analysis?  
Thanks a lot!    
 A: With $a,m,n \in \mathbb{R}$, the taxonomy from Hurkyl's answer in the question linked by Eli Rose and avoiding complex intermediates and exponentiation, the values that make the identity true are:
Case $a>0$: no restrictions on $m,n$.
Case $a=0$: requires $m,n \ge 0$ if you allow/define $0^0 \in \mathbb{R}$ (it depends on convenience and context), $m,n > 0$ otherwise.
Case $a<0$: The continuous real exponentiation operator forbids $a < 0$. The discrete real exponentiation operator (which allows $x^{1/3}$ etc. to be defined over all $\mathbb{R}$) requires $m=\dfrac{p_m}{q_m} \in \mathbb{Q}$ with $q_m$ odd (includes any $m \in \mathbb{Z}$ since then $m=\dfrac{m}{1}$). Similarly, the RHS of the identity requires $mn = \dfrac{p_{mn}}{q_{mn}} \in \mathbb{Q}$ with $q_{mn}$ odd. Then $n = \dfrac{mn}{m} = \dfrac{p_{mn}q_m}{q_{mn}p_m}$. If $p_m$ is odd, $a^m < 0$ and we require the denominator of $n$ to be odd, and this is the case since $q_{mn}, p_m$ are odd by assumption. If $p_m$ is even, $a^m > 0$ and there are no further constraints on $n$.
The most compact expression I can devise, if you like $x^{1/3}$ etc. defined over all $\mathbb{R}$, is: $(a^m)^n = a^{mn}$ is true/meaningful for $a,m,n \in \mathbb{R}$ if $a > 0$, or if $a = 0$ and $m,n > 0$ (or $\ge 0$ if $0^0$ defined), or if $a < 0$ and $m,n \in\mathbb{Q}$ with the denominators of $m$ and $mn$ odd in lowest terms.
I've had no luck finding book-based references to the $a<0$ case. The analysis and calculus texts I have at hand deal with the case $a>0$.
Good elementary algebra texts have the rule that only odd (integral) roots of negative numbers are meaningful and give rules like $\sqrt[q]{a^x} = a^{x/q}$, but I haven't found a clear treatment anywhere.
