It depends on what you can use, of course. Let's say you need to find the square root of a number x
A first, but very rough way, consists in simply finding two numbers, n and m, such that n^2 < x < m^2. If you have this relation, you can surely affirm that \sqrt{x} is some number between n and m. For example, if we needed to estimate \sqrt{40}, we could say that it surely is a number between 6 and 7.
Of course, this method can be used also with rational numbers. For example, with a bit of calculations you can find out that \sqrt{40} actually lies between 6.3 and 6.4, and so on.
Another way could be factoring x with primes, and simplify squared factors, if any appear. This could leave only smaller roots to calculate: consider for example \sqrt{18}. You can write 18 as 2\cdot 3^2, and so \sqrt{18}=\sqrt{2\cdot 3^2}=\sqrt{2}\cdot\sqrt{3^2}=3\sqrt{2}
