Squares and Square Roots
First learn about Squares, then Square Roots are easy.
How to Square A Number
To square a number, just multiply it by itself ...
Example: What is 3 squared?
3 Squared  =  = 3 × 3 = 9 
"Squared" is often written as a little 2 like this:
This says "4 Squared equals 16"
(the little 2 says
the number appears twice in multiplying)
Squares From 0^{2} to 6^{2}
0 Squared  =  0^{2}  =  0 × 0  =  0 
1 Squared  =  1^{2}  =  1 × 1  =  1 
2 Squared  =  2^{2}  =  2 × 2  =  4 
3 Squared  =  3^{2}  =  3 × 3  =  9 
4 Squared  =  4^{2}  =  4 × 4  =  16 
5 Squared  =  5^{2}  =  5 × 5  =  25 
6 Squared  =  6^{2}  =  6 × 6  =  36 
The squares are also on the Multiplication Table: 
Negative Numbers
We can also square negative numbers.
Example: What happens when we square (−5) ?
Answer:
(−5) × (−5) = 25
(because a negative times a negative gives a positive)
That was interesting!
When we square a negative number we get a positive result.
Just the same as squaring a positive number:
(For more detail read Squares and Square Roots in Algebra)
Square Roots
A square root goes the other way:
3 squared is 9, so a square root of 9 is 3
A square root of a number is ...
A square root of 9 is ...
It is like asking:
What can we multiply by itself to get this?
To help you remember think of the root of a tree: "I know the tree, but what root made 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 
Decimal Numbers
It also works for decimal numbers.
Try the sliders below. Note: the numbers here are only shown to 2 decimal places.
Using the sliders (remembering it is only accurate to 2 decimal places):
 What is the square root of 8?
 What is the square root of 9?
 What is the square root of 10?
 What is 1 squared?
 What is 1.1 squared?
 What is 2.6 squared?
Negatives
We discovered earlier that we can square negative numbers:
Example: (−3) squared
(−3) × (−3) = 9
And of course 3 × 3 = 9 also.
So the square root of 9 could be −3 or +3
Example: What are the square roots of 25?
(−5) × (−5) = 25
5 × 5 = 25
So the square roots of 25 are −5 and +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 mathematics look important! 
We use it like this:
and we say "square root of 9 equals 3"
Example: What is √25?
Well, we just happen to know that 25 = 5 × 5, so when we multiply 5 by itself (5 × 5) we will get 25.
So the answer is:
√25 = 5
But wait a minute! Can't the square root also be −5? Because (−5) × (−5) = 25 too.
 Well the square root of 25 could be −5 or +5.
 But when we use the radical symbol √ we only give the positive (or zero) result.
Example: What is √36 ?
Answer: 6 × 6 = 36, so √36 = 6
Perfect Squares
The Perfect Squares (also called "Square Numbers") are the squares of the integers:
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  etc  
Perfect Squares:  0  1  4  9  16  25  36  49  64  81  100  121  144  169  196  225  ... 
Try to remember them up to 12.
Calculating Square Roots
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 √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
 ...
Getting closer to 10, but it will take a long time to get a good answer!
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 ! 
Note: numbers like that are called Irrational Numbers, if you want to know more.
The Easiest Way to Calculate a Square Root
Use your calculator's square root button! 
And also use your common sense to make sure you have the right answer.
A Fun Way to Calculate a Square Root
There is a fun method for calculating a square root that gets more and more accurate each time around:
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 (4 + 2.5 = 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 
 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
Now ... why don't you try calculating the square root of 2 this way?
How to Guess
What if we have to guess the square root for a difficult number such as "82,163" ... ?
In that case we could think "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 is a good start.
Square Root Day
The 4th of April 2016 is a Square Root Day, because the date looks like 4/4/16
The next after that is the 5th of May 2025 (5/5/25)