What is the value of $(-1)^{\frac {2}{2}}$?

It depends on what the person who wrote down $(-1)^{\frac22}$ meant by writing that.

The notation $A^B$ has a well-defined meaning that everybody agree on in the following cases

• $B$ is a natural number.
• $B$ is a (possibly negative) integer, and $A\ne 0$.
• $B$ is real and $A$ is positive.
• $A$ is $e$, and $B$ is any complex number.

The definitions of exponentiation that work in each of these cases are different, but fortunately they agree about the result whenever some combination of $A$ and $B$ fits into more than one of them.

Outside of these cases, ambiguity creeps in. In the general case of complex $A$ and $B$, one can treat $A^B$ as a multi-valued expression, but in some contexts it can be useful to single out a single one of the multiple values as the one we mean. (Indeed this is what happens in the two last of the cases above, where the canonical meaning of $A^B$ is only one of the multiple values the general case allows).

In particular, the rule that $A^{BC} = (A^B)^C$ holds in the three first cases cases, but not necessarily for a wider generalization that one might wish to use in a particular case.

For your particular case $(-1)^{\frac 22}$ does not immediately fall into any of the above categories. If we observe that $\frac22=1$, then we can rewrite it to $(-1)^1$ which does match the first two cases and gives the result $-1$.

However, the fact that somebody wrote the expression as $(-1)^{\frac22}$ suggests that they may have meant something different from that, and then there's no way out of either asking them, or try to figure out for yourself which interpretation of the notation makes sense in the context you see it in.