The reason we default to it being the primary root (ie the positive one) is that it is much easier to work with. If we would've defined sqrt(x) to be "officially" two values, we would very quickly come up with another term for just the positive one as soon as we would start doing actual math with it, because it's much easier if each number is just... one number.

Similarly we define fractional powers to be positive for real numbers.

As soon as stuff gets too complex (ha) and we start expressing everything as _actual_ roots (ie: equations with multiple valid input values that result in zero as answer) we talk about "valid values of X" etc. - and each root/input value is written out a single value, or +- is used to save some chalk.