It's been several years since high school. How to you find the square root of a number?

Why exactly do you need this?

Virtually all calculators produced today have a square root function. Just use that.
If you aren't allowed to use a calculator for whatever reason, you can always use the guess and check method.
i.e. starting with, say, 45. You know that 6^2=36, and 7^2=49, so you can reasonably guess that the square root of 45 is ~6.7
Or, you can do it the LONG way...
It's too lenghty to type out here (for me at least, :p) so I'll just link you a good explaination:
How to manually find a square root (http://math.arizona.edu/~kerl/doc/square-root.html)

Here is an iterative method I prefer if I have to do these by hand. It is called the Babylonian method.

Say we want the square root of 14, \sqrt{14}

Make a guess for the first iteration. We know that it is going to be between 3 and 4, because 3^2=9 and 4^2=16.

The formula is x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{S}{x_{n}} \right)

where S is the number we are finding the square root of.

Let's make a guess of x_{0}=3.5

\frac{1}{2}\left(3.5+\frac{14}{3.5}\right)=\fbox{3 .75}

Next iteration:

\frac{1}{2}\left(3.75+\frac{14}{3.75}\right)=\fbox {3.741667}

Next iteration:

\frac{1}{2}\left(3.741667+\frac{14}{3.741667} \right)=\fbox{3.74165738679}

Compare to \sqrt{14}=3.7416573867739

It only took 3 iterations to get it accurate to within about 10 decimals places.

The better the initial guess, the faster it will converge. You can even start with a bad guess and it will converge, though not as fast.

I'll explain how the formula is good, if you allow me.

You want to know:

\sqrt{14} = x

Let x be the square root.

Then, square both sides:

14 = x^2

Since it's an equation, you can add x^2 on both sides:

14 + x^2 = x^2 + x^2

14 + x^2 = 2x^2

Now divide by 2x;

\frac{14 + x^2}{2x} = \frac{2x^2}{2x}

\frac{14}{2x} + \frac{x^2}{2x} = x

\frac12(\frac{14}{x} + x) = x

x =\frac12(x+\frac{14}{x} )

The square root of a number =

Say the square root of :-1234567890.=35136 See the workings below.
This is very hard to portray on this medium. The trick is the side figures are multuplied by 2. e.g 35*2=70, 351*2 = 702
3513*2 = 7026

3 5 1 3 6
/12,34,56,78,90
3 9
---------------------------------
3,34
65 3,25
--------------------------------
9,56
701 7,01
------------------------------
2,55,78
7023 2,10,69
--------------------------------------
45,09,90
70236 42,14,16
-----------------------------------------
2,95,74
========================

The square root of 12,34,56,78,90 os apprximately 35136.

Yes, maybe you could try using the
tabs so that the spaces are kept and that it becomes easier to understand.

Like this:

[code]
3 5 1 3 6
/12,34,56,78,90
3 9
------------------------
3,34
65 3,25
------------------------
9,56
701 7,01
------------------------
2,55,78
7023 2,10,69
------------------------
45,09,90
70236 42,14,16
------------------------
2,95,74
========================

But I'm not sure if the spacings are good... =/ You might want you re-adjust them.