Why doesn't $z^n\cdot\left(\frac{a+b}{z}\right)^n = (a+b)^n$ always hold? When I entered is(z^n*((a+b)/z)^n = (a+b)^n); into Maple, the output was false and I guess Maple assumes that $a,b,n$ and $z$ can be any number in $ℂ$.
I thought
$$z^n\cdot\left(\frac{a+b}{z}\right)^n$$
was clearly
$$\left(z\cdot\frac{a+b}{z}\right)^n$$
and therefore
$$(a+b)^n$$
Why does this not apply?
Edit:
The output is still false when typing assume(z <> 0): is(z^n*((a+b)/z)^n = (a+b)^n);
Edit 2:
assume(z >= 0): is(z^n*((a+b)/z)^n = (a+b)^n); returns true.
Edit 3:
assume(z = 0): is(z^n*((a+b)/z)^n = (a+b)^n); returns false.  
Isn't this a contradiction?
Edit 4:
assume(n in ℕ): is(z^n*((a+b)/z)^n = (a+b)^n); returns false.
For $a<0$ and $z<0$, Maple fails to evaluate the expression, although it is true. This means Maple returned false instead of admitting that it is unable to determine the result.
This is a flaw.
 A: In complex numbers, $(ab)^c=a^cb^c$ does not necessarily hold.
$$((-1)(-1))^{1/2}=1\ne(-1)^{1/2}(-1)^{1/2}=i^2=-1.$$
A: It looks like Maple has trouble evaluating the left hand side of your equation. You can help Maple by asking it to expand (or simplify) the left-hand-side of your expression before using the is command from the assume facility. 
eqn := z^n*((a+b)/z)^n = (a+b)^n;
assume(z::real,n::integer);

As we saw in your question, is(eqn); returns false but
is(simplify(eqn));

returns true. I suppose this arises from limitations of the Assume Facility; as referred to on the help page, this may be documented in [Corless, Monagan] (I was unable to find an online copy of it). 

Side note
If $n$ is any complex number, we cannot say anything in general (non-uniqueness) but applying the assumption that $(a+b)/z>0$ seems to be enough for the Assume Facility (with a helping hand from the simplify command):
restart;
eqn := z^n*((a+b)/z)^n = (a+b)^n;

assume((a+b)/z>1);
is(simplify(eqn));

returns true although $z^n$ is ambiguous for $z,n \in \mathbb C$. This seems to be a result of the way Maple defines the complex exponential to have a unique solution, see Section 5.1 The Complex Exponential Function (document can only be opened in Maple).
[Corless, Monagan] Corless, Robert, and Monagan, Michael. "Simplification and the Assume Facility." Maple Technical Newsletter, Vol. 1 No. 1. Birkhauser, 1994. 
