Why define nth root as supremum of a set?

Question

Tao's Analysis I says:

Why would we do this instead of saying $x^{1/n}$ is a real $y$ such that $y^n=x$?

Answer 1

Well, how do you know that there exists $y$ such that $y^n=x$, or that there is a unique such $y$? This is not obvious (and in fact such a $y$ is not unique if $n$ is even so you should require $y\geq 0$), and until you have proven this your definition does not necessarily define a specific real number at all. On the other hand, Tao's definition clearly does define a real number (all you have to verify is that the set you are taking the supremum of is nonempty and bounded above). So you can refer to $x^{1/n}$ immediately after defining it without doing a lot of work to prove your definition is meaningful, which is convenient.