*Originally posted by piderman*

**Care to explain how it works (and the mechanics behind it)? Say I want to take the square root of 5764801 ðŸ˜‰ðŸ˜€**

The mechanics on how it works is not too hard to see once you see the method. It's the method that is hard to type down :p

OK, the process is supposed to look like a division process.

devide the number in parts of two, starting on the right;

5 76 48 01

We start with the 5. The biggest square smaller then 5 is 2*2, so the first digit of the root is a TWO

5-4 = 1 and we take the next two digits of the square making 176

Now we had 2*2, instead of multiplying we add the two two's now, giving a 4. Now we need to find x such that (40+x)*x is as close under 176 as possible.

In this situation x = 4 since 44*4=176. So the second digit of our root is FOUR

176-176=0 and we add the next two digits of our square to make 48

We had 44*4, and now instead of multiplying we add the two numbers to make 44+4=48. Now we need to find x so that (480+x)*x is as close under 48 as possible.

480*0=0 so the third digit of our root is ZERO.

48-0 = 48 and adding the last two digits of our sqaure to it gives 4801.

480+0=480, so now we need to find x so that (4800+x)*x is as close under 4801 as possible. This time x=1 and this leaves remainder 0, meaning we're done. The last digit of our root is 1

So sqrt(5764801)=2401.

The idea behind this method is that you start with a square of size 5, and take a square out of it (here size 4). What remains is an edge consisting of a square and two rectangles. Now you increase the size of the sides by a factor 10 (this is adding the next two digits of our square, as the square we had increases by a factor 100).

Now we try to find x such that 20x + 20x + x^2 approximates the edge as close as possible. above can be rewritten as (40+x)*x wich is the formula we solved.

You keep getting edges as remainder, and you keep approximating it as close as possible, increasing the edges by a factor 10 each time, until you're done.

I hope this made sense, it's much easier to explain on paper ðŸ™‚