I have spent many hours trying to find out by myself, the following:
Specifically I mean the relationship between dividing a number "n", between p, and getting the root of "n" with an index "p", then what is the algebraic reason between both operations?
Besides, which branch of mathematics is studying this?
You are looking for the Big O Notation, which "describes the limiting behaviour of a function when the argument tend towards a particular value or infinity".
Square root is $\mathcal{O}(x^{1/p})$ and division by a constant is $\mathcal{O}(x)$, hence their growth rates are clearly compared in this way.