When given a language, does the algebra follow when raising a symbol in the alphabet to 0 or a negative number? For example, I'm given the language {$a^j b a^k | j < k + 4$}. Do I have to worry about cases like $a^0$? Does it come out to 1 or $\epsilon$? What about things like $a^{-1}$?

$a^0$ means the empty word $\epsilon$, though it's not sure whether they allowed $j=0$ or not.
Negative powers are not defined in this setting, the words only form a semigroup (actually, monoid with $\epsilon$), not a group, so that we don't have inverses. 
