Is it possible to find square root using only rational numbers and elementary arithmetic operators Suppose I have a number a
How can I find it's square root using only +, -, /, * and rational numbers?
If it is impossible how to prove it?

If you allow infinite number of operations, then you can use some algorithm.
One easy example is root searching via Newton's method. Here we do the iteration
$$x_{n+1} = \frac{a + x_n^2}{2x_n},$$
which eventually converges to $\sqrt{a}$ if $a$ and $x_0$ are positive.
See https://en.wikipedia.org/wiki/Methods_of_computing_square_roots for other methods.
