They are fundamental to the arithmetic used in analysis and algebra.

One can get along quite well without them once you replace the arithmetic expressions with more general objects.

Just come on. The square root of 2 is the easiest example of an irrational number, this has been known since Ancient Greece. You can't compute distances in Euclidean spaces without the square root. "Solving equations by roots" is the bread and butter of algebra. Adjoining roots to a field is how you get Galois Theory. Several algorithms related to number theory have complexity O(sqrt(n)). And so on.

You chose an extremely poor example and now you're trying to die on that hill. Please don't die on that hill.

Also fundamental to geometry (pythagorean equation and others).