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:
r2 = x
r is the square root
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 "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 r2 = x
Answer: √2 × √2 = 2
That last example is there to show you how the definition r2 = 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:
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 w2 = 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.
Plus-Minus 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: | r2 = 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 x2-9 = 0
 |
Start with: x2-9 = 0 Move 9 to right: x2 = 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: (x-3)2 = 16
Start with: (x-3)2 = 16
Take Square Root: x-3 = ±√16 = ±4
Move 3 to the right: x = 3±4
Answer: x = 7 or -1
Check: (7-3)2 = 42 = 16
Check: (-1-3)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 one-half:
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.