does the square root of 256 represent the positive square root of 256, the negative square root of 256, or both?
As long as you're in the realm of mathematics, it represents both. If it arises as the solution to a real-world situation, then one of the solutions may have no practical application, and you'll discard it because it's not a helpful one.
Both, because a negative squared and a positive squared would both still give you a positive product. For example: -4 x -4 = 16 and 4 x 4 = 16 too.