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}#
