I have a hypothetical question: suppose there exists an algorithm that solves an NP-Complete problem polynomial time, but requires the computation
of values that grow exponentially big (or small). 

For example, suppose POLY-3SAT solves 3-SAT in N^17, however to do so it must compute/evaluate the value of a number C whose value grows as N^1000 (or 1/N^1000).

What would this imply? Does computing/evaluating an exponentially large value automatically place the algorithm in EXP-SPACE (or some other complexity class)? It seems like this would be a different complexity class than just ordinary P.