Question 1067437
I would answer by convention the index is the number of roots taken, and conventionally we don't have a way of multiplying a number by itself 1.5 times or -1 times. 
Another concern is when we get into logs, log2 (4)=2; we don't have negative bases (yes, in complex math they can occur, but in the math that nearly everybody else does, they cannot be.) While -2^2=4 we can't say log (-2)4=2; they aren't defined for negative numbers.
While the 2 means raising the power to (1/2) and the 3 to (1/3) making it a (-1/2) doesn't raise it to the -2 power.
Some things are simply defined, like 0!, which is defined as 1.  All the factorials I have ever looked at have never been negative, but there does exist a -1!, but you are very much in the realm of very complex mathematics where few of us ever go.