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
".