Exponentiation does have some scenarios where it can be defined without an (obvious) exponential function, and roots may not be uniquely defined (or rather they

Exponentiation does have some scenarios where it can be defined without an (obvious) exponential function, and roots may not be uniquely defined (or rather they are almost never unique), which means you need to be a bit careful, and which means that in theory the 3rd root could differ from the definition of x^(1/3). However in the cases where you need to be careful most of stuff you'd use the more general notation for wouldn't be applicable anyway, you'd have a high chance of writing down an expression that has no unique value, or can't even be evaluated.