I have been pondering the following question: when considering perfect roots of a number, what are the smallest and largest values one should consider?
Obviously, the smallest to consider is 2, there is no point in considering the 1st root of a number. But is there really a largest value for any number to consider when looking for a perfect root?
Say for example we take the number 10: the smallest perfect root to consider is 2. But what would you say the largest is?
If the number is $n$, the largest is $\lfloor \sqrt n \rfloor$. If the square of that is $n$, you have found a perfect root. No perfect root can be larger. For your example, $\lfloor \sqrt {10} \rfloor=3$ and when we see $3^2=9 \neq 10$ we are done.