An expression or function evaluation has only one value (or "answer") - or perhaps none in case it is not defined. Even though the equation $x^2=9$ has two real solutions $x=3$ and $x=-3$, the expression $\sqrt 9$ has only one value $3$.
What "is" $a^b$, anyway? In order to know what is meant by $a^b$, we need to define the expression $a^b$ for suitable $a$ and $b$.
Unfortunately, we don't do this all at once, but first only for a few cases and then extend the definition. And even more unfortunately, sometimes we encounter different routes to extend along and have to keep an eye on consistency. Here's an overview.
For
non-negative integers $a,b$,
i.e., when these can be interpreted as the cardinalities of finite sets, we can readily define $a^b$ as the number of maps from a set $B$ of cardinality $b$ to a set $A$ of cardinality $a$. For example, simple counting then gives us $3^2=9$, $2^3=8$, $0^0=1$, and so on. One can even show that this implies
$$\tag1 a^0=1,\quad a^{n+1}=a\cdot a^n$$
and more general relations such as
$$\tag2 a^{b+c}=a^ba^b,\quad a^{bc}={(a^b)}^c,\quad (ab)^c=a^cb^c.$$
We can use $(1)$ to extend our definition (namely, by using recursion as definition) to the case
$a$ an arbitrary number, $b$ a non-negative integer.
When $a\ne0$, we can even use the recursion in $(1)$ to go backwards and thereby extend to
$a\ne0$, $b\in\Bbb Z$.
Note that we cannot nicely define $0^b$ when $b$ is a negative integer. Well, theoretically we can define anything - but a definition should be useful. For example, we want to be able to make use of properties $(2)$ without having to check for applicability. Thus any extension of our definition should make sure that $(2)$ keeps holding. This is called the Principle of Permanence. Therefore any useful definition of $0^{-n}$ should make $0=0^{-n}\cdot 0=0^{-n}\cdot 0^n=0^0=1$, which is absurd.
But can we extend to rational exponents $b=\frac nm$, $n\in \Bbb Z$, $m\in \Bbb Z_{>0}$ (respecting the Principle of Permanence)?
That is, $x=a^{\frac nm}$ should be a number with the property $x^m=a^n$. Sometimes there is no (real) number with this property, (namely when $a<0$, $m$ even, $n$ odd; or when $a=0$ and $\frac nm<0$). Sometimes there is exactly one real solution (namely when $m$ is odd; also when $a=0$ and $\frac nm>0$). And sometimes there are several solutions (namely when $a>0$, $m$ even, $n$ odd). In the latter case we have to remember that we want to define the value of $a^b$ as a function of $a$ and $b$. The convention is to take the positive value, which has a huge advantage when we extend even further. But first let's summarize that we managed to deal with
$b\in\Bbb Q$, provided $a>0$, or $a=0$ and $b\ge0$, or $a<0$ and denominator of $b$ is odd.
Apparently, things are already becoming somewhat convoluted.
Thanks to our choice odf positive solution above, we have a new property: If $a>0$, then the map $\Bbb Q\to\Bbb R$, $b\mapsto a^b$ is continuous! This allows us to extend to
$a>0$, $b\in\Bbb R$
simply by continuity! This cannot work for negative $a$ because of the gaps at rationals with even denominator. And it almost works for $a=0$ by defining $0^b=0$ for all $b>0$, but neither can we define this for negative exponents, nor can we avoid the discontinuity at $b=0$, i.e., $0^0=1\ne 0=\lim_{b\to0^+}0^b$.
Once we know enough calculus, we can use a different way to extend our definition by setting $a^b=\exp(b\ln a)$ using the exponential function and natural logarithm. This agrees with the above as long as $a>0$ (in particular, we still have $4^{\frac12}=2$, period). It cannot deal with the case of $a=0$ where the logarithm is not defined and also has serious trouble when $a<0$. On the other hand, it allows us to use complex exponents $b$.
The goal now seems to be to extend the definition of the $\ln$ function. There are nice was to do so for any open subset $U$ of $\Bbb C$ that is a simply connected neighbourhood of $1$. While there is a somewhat natural choice $U=\Bbb C\setminus (-\infty,0]$ for a maximal such $U$, different choices may sometimes be more useful (and do ultimately lead to different values for some $a^b$!). Note that this standard choice (the "principal branch") ignores the cases with $a<0$, and just combining the two methods is also no good idea regarding Permanence.
to indicate you want the principal root, how should that be indicated? – Randor Feb 20 '21 at 10:54