Squares and Square Roots
To understand square roots, first you must understand
squares ...
How to Square A Number
To square a number, just multiply it by itself
...
Example: What is 3 squared?
3 Squared 
= 

= 
3 × 3 
= 
9 
Note: we write down "3
Squared" as 3^{2}
(the little "^{2}" means
the number appears twice in multiplying)
Some More Squares
4 Squared 
= 
4^{2} 
= 
4 × 4 
= 
16 
5 Squared 
= 
5^{2} 
= 
5 × 5 
= 
25 
6 Squared 
= 
6^{2} 
= 
6 × 6 
= 
36 
Square Root
A square root goes the other direction:
3 squared is 9, so the square root
of 9 is 3
3 

9 
The square root of a number is ...
... that special value that when multiplied by itself gives the original number.
The square root of 9 is ...
... 3, because when 3 is multiplied by itself you get 9.

Note: When you see "root" think
"I know the tree, but what is the root that produced it?"
In this case the tree is "9", and the root is "3". 
Here are some more squares and square roots:

4 

16 
5 

25 
6 

36 
Example: What is the square root of 25?
Well, we just happen to know that 25 = 5 × 5, so if you multiply
5 by itself (5 × 5) you will get 25.
So the answer is 5
The Square Root Symbol

This is the special symbol that means "square root",
it is sort of like a tick, and actually started hundreds of years
ago as a dot with a flick upwards.
It is called the radical, and always makes math look important! 
You can use it like this: (you would say "the square root of 9 equals 3")
More Advanced Topics Follow
You Can Also Square Negative Numbers
Have a look at this:
So the square root of 25 can be 5 or 5
There can be a positive or negative answer to a square root!
But when people talk about "the" square root they usually mean just the positive one.
And when you use the radical symbol √ it always means just the positive one.
Example:
A square root of 36 could be 6 or 6, but √36 = 6 (not 6)
Perfect Squares
The perfect squares are the squares of the whole numbers:

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
etc 
Perfect Squares: 
1 
4 
9 
16 
25 
36 
49 
64 
81 
100 
121 
144 
169 
196 
225 
... 
Square Roots of Other Numbers
It is easy to work out the square root of a perfect square, but it
is really hard to work out other square roots.
Example: what is the square root of 10?
Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.
 Let's try 3.5: 3.5 × 3.5 = 12.25
 Let's try 3.2: 3.2 × 3.2 = 10.24
 Let's try 3.1: 3.1 × 3.1 = 9.61
Very slow ... at this point, I get out my calculator and it says:
3.1622776601683793319988935444327
... but the digits just go on and on, without any pattern. So even
the calculator's answer is only an approximation !
(Further reading: those kind of numbers are called surds which are a special type of irrational number)
A Special Method for Calculating a Square Root
There are many ways to calculate a square root, but my favorite method is an easy one which gets more and more accurate depending on how many times you use it:
a) start with a guess (let's guess 4 is the square root of 10)
b) divide by the guess (10/4 = 2.5)
c) add that to the guess (2.5+4=6.5)
d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)
e) now, set that as the new guess, and start at b) again
... so, our first attempt got us from 4 to 3.25
Going again (b to e) gets us: 3.163
Going again (b to e) gets us: 3.1623
And so, after 3 times around the answer is 3.1623, which is pretty good, because:
3.1623 x 3.1623 = 10.00014
This is a fun method to try  why not use it to try calculating the square root of 2?
How to Guess
What if you have to guess the square root for a difficult number such as "82,163" ... ?
In that case I would think to myself "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 would be a good start.
