Squares and Square Roots in Algebra
You might like to read our Introduction to Squares and Square Roots first.
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, so 4×4=16)
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, 
We can use it like this:
we say "square root of 9 equals 3"
Example: What is √36 ?
Answer: 6 × 6 = 36, so √36 = 6
Negative Numbers
We can also square negative numbers.
Example: What is minus 5 squared?
But hang on ... what does "minus 5 squared" mean?
 square the 5, then do the minus?
 Or square (−5)?
It isn't clear! And we get different answers:
 Square the 5, then do the minus: −(5×5) = −25
 Square (−5): (−5)×(−5) = +25
So let's make it clear by using "( )".
Example Corrected: What is (−5)^{2} ?
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 when we square a positive number:
Remember our defintion of a square root?
r^{2} = x
r is the square root
And we just found that:
(−5)^{2} = 25
(+5)^{2} = 25
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 we 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 
The "±" tells us to include the "−3" answer also.
Example: Solve for x in (x − 3)^{2} = 16
Start with:  (x − 3)^{2} = 16  
Take Square Root:  x − 3 = ±√16  
Calculate √16:  x − 3 = ±4  
Move 3 to the right:  x = 3 ± 4  
Answer:  x = 7 or −1 
Check: (7−3)^{2} = 4^{2} = 16
Check: (−1−3)^{2} = (−4)^{2} = 16
Square Root of xy
When two numbers are multiplied within a square root, we can split it into a multiplication of two square roots like this:
√xy = √x√y
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 
And √x√y = √xy :
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!
We can use Imaginary Numbers, but still get the wrong answer of −4
Oh that's right ...
The rule only works when x and y are both greater than or equal to 0
So we can't use that rule here.
Instead just do it this way:
√(−8 × −2) = √16 = +4
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 result is an Imaginary Number... read that page to learn more.