Squares and Square Roots in Algebra
You might like to read our Introduction to Squares and Square Roots first, but here is a quick summary:
Squares
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 means
the number appears twice in multiplying)
Square Root
A square root goes the other direction:
3 squared is 9, so a square root of 9 is 3
It is like asking:
What can I multiply by itself to get this?
Definition
Here is the definition:
A square root of x is a number whose square is x:
r^{2} = x
r is the square root
The Square Root Symbol
This is the special symbol that means "square root",
it is like a tick, 
You can use it like this:
you would say "square root of 9 equals 3"
Example: What is √36 ?
Answer: 6 × 6 = 36, so √36 = 6
Example: What is √2 × √2 ?
Remember the definition: The square root of x is "r" where r^{2} = x
Answer: √2 × √2 = 2
That last example is there to show you how the definition r^{2} = x works.
Negative Numbers
You can also square negative numbers.
Example: What is (5)^{2} ?
Answer:
(5) × (5) = 25
(because a negative times a negative gives a positive)
That was interesting!
When you square a negative number you get a positive result.
Just the same as if you had squared a positive number:
Be careful ... when someone says "minus 5 squared" do we:
 Square the 5, then do the minus?
 Or do we square (−5) ?
It isn't clear! We can get different answers:
 Square the 5, then do the minus: −(5×5) = 25
 Or do we square (−5): (−5)×(−5) = +25
So make it clear by using the "( )".
Two Square Roots
And that means ...
... a square root of 25 can be 5 or 5
So there can be a positive or negative square root!
This is important to remember!
Example: Solve w^{2} = a
Answer:
w = √a or w = √a
Principal Square Root
So if there are really two square roots, why do people say √25 = 5 ?
Because √ means the principal square root ... the one that isn't negative!
There are two square roots, but the symbol √ means just the principal square root.
Example:
The square roots of 36 are 6 and 6
But √36 = 6 (not 6)
The Principal Square Root is sometimes called the Positive Square Root.
PlusMinus Sign
±  is a special symbol that means "plus or minus", 
so instead of writing:  w = √a or w = √a  
we can write:  w = ±√a 
In a Nutshell
When we have:  r^{2} = x 

then:  r = ±√x 
Why Is This Important?
Why is this "plus or minus" Important? Because you don't want to miss a solution!
Example: Solve x^{2}9 = 0
Start with: x^{2}9 = 0 Move 9 to right: x^{2} = 9 Take Square Root: x = ±√9 Answer: x = ±3 
If we don't remember the "±" we would miss the "3" answer
Example: Solve for x: (x3)^{2} = 16
Start with: (x3)^{2} = 16
Take Square Root: x3 = ±√16 = ±4
Move 3 to the right: x = 3±4
Answer: x = 7 or 1
Check: (73)^{2} = 4^{2} = 16
Check: (13)^{2} = (4)^{2} = 16
Square Root of xy
When two numbers are multiplied within a square root, you can split it into a multiplication of two square roots like this:
but only when x and y are both greater than or equal to 0
Example: What is √(100×4) ?
√(100×4)  = √(100) × √(4) 
= 10 × 2  
= 20 
Example: What is √8√2 ?
√8√2  = √(8×2) 
= √16  
= 4 
Example: What is √(8 × 2) ?
√(8 × 2)  = √(8) × √(2) 
= ??? 
We seem to have fallen into some trap here!
(If I continued this I would need to use Imaginary Numbers,
and the answer would be 4, even though √(8 × 2) = √16 = +4)
Oh that's right ...
The rule only works when x and y are both greater than or equal to 0
An Exponent of a Half
A square root can also be written as a fractional exponent of onehalf:
but only for x greater than or equal to 0
How About the Square Root of Negatives?
The answer will be an Imaginary Number... read that page to learn more.