ThomasPierson: If a square root must be used to represent an approximation then the concept of accuracy ceases to be absolute and becomes relative.
Actually it's exactly the opposite :-) Square root of a given number is an exact value. Only if you try to calculate it and represent as a decimal number you get an approximation.
As for why it exists in the first place - if you like a more philosophical answer, it's a link between stuff in two dimensions (surface of a square) and stuff in one dimension (side of a square). If you want to go from three dimensions to one (cube volume to cube height) you use a cube root, and so on :-)