I just saw a solution that says it is since, for any complex number $z$
$$1^z = e^{z\log1} = e^{z(0)} = 1$$
However, isn't this only true for the principle branch of $1^z$, since by definiton, (letting capital L denote the princple value of log
$$\log1 = \operatorname{Log}1 + i2\pi k$$
for any $k \in \mathbb{Z}$?