# How do you estimate square roots?

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 $\setminus \sqrt{x}$ is some number between $n$ and $m$. For example, if we needed to estimate $\setminus \sqrt{40}$, 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 $\setminus \sqrt{40}$ 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 $\setminus \sqrt{18}$. You can write $18$ as $2 \setminus \cdot {3}^{2}$, and so $\setminus \sqrt{18} = \setminus \sqrt{2 \setminus \cdot {3}^{2}} = \setminus \sqrt{2} \setminus \cdot \setminus \sqrt{{3}^{2}} = 3 \setminus \sqrt{2}$