I myself teach that when we do (nonpositive number)^(noninteger number) we enter a minefield, where familiar rules break down. It seems that it is not widely taught.

You really only need that b is real.

In[120]:= e1 = a^(b c);
e2 = (a^b)^c;

In[122]:= Simplify[e1 - e2, Assumptions -> {a > 0, Element[b, Reals]}]

Out[122]= 0

By definition a^b is Exp[b*Log[a]]. From there it can be worked out that the value of c is not relevant in this, and it comes down to whether Log[a^b] == Log[a]*b. The conditions shown above are sufficient though not necessary.

Possibly someone has a decent reference handy. Mine is "I worked it out by hand using the definition of a^b".

Unfortunately this is not taught well in US high schools. A related subject is that students lean that (-8.0)^(1/3) = -2.0. However, Mathematica and most software that will do that computation give a complex number. I think high schools should cover (negative)^(1/2) and (negative)^(integer) but not mention (negative)^x. I also think all high school students should lean the content in the attached.

Attachments:

Properties of Ex...pdf

Actually we only need the following:

Simplify[(a^b)^c == a^(b c), Element[c, Integers]] 

That and much more is in the document I uploaded above. A slightly obscure case that I didn't include in my attachment is the following

Simplify[(a^b)^c==a^(b c),-1<b<1]

Both of the above are true for all complex numbers except when the second argument of Simplify specifies otherwise.