How can I find the square root using pen and paper? Okay, I know this is very basic question. I learned 2 methods in school. But now, I forget one. 
Here is a simple method that I know.


*

*Find the prime divisors of the number

*Omit the half of numbers that have been appeared even times

*multiply the rest


For example you want to find square root of 36. You find the divisors. They are 2x2x3x3. In step 2 they appeared as 2x3. That is 6. Problem is this method works when the square root is an integer number. It doesn't work for numbers that doesn't have integer square root. Like 38.
So my question is how can I find the square root of any arbitrary number using pen and paper?

If your number is close enough to a perfect square, you can use the expansion
$$
\sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 +- \cdots + (-1)^{n-1}\frac{(2n-3)!!}{2\cdot(2n)!!}x^n+\cdots
$$
where
$$
(2n)!! = (2n)(2n-2)\cdots4\cdot2,\quad(2n+1)!!=(2n+1)(2n-1)\cdots3\cdot1.
$$
A famous physist R. Feynman is said to have used this formula and beat an abacus. 
