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 = 3x3 box = 3 × 3 = 9

"Squared" is often written as a little 2 like this:

4 Squared equals 16
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:

square root of 9 is 3

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

radical symbol  

This is the special symbol that means "square root", it is 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 can use it like this:

square root of 9
we say "square root of 9 equals 3"

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

And an example to show how the definition r2 = x works:

Example: What is √2 × √2 ?

Remember the definition: The square root of x is "r" where r2 = x

The square root of 2 is "r" where r2 = 2
√2 is "r" where r2 = 2
(√2)2 = 2
√2 × √2 = 2

Answer: √2 × √2 = 2

Negative Numbers

We 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 we square a negative number we get a positive result.

Just the same as when we square a positive number:

5x5 = -5x-5

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 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 we don't want to miss a solution!

Example: Solve x2 − 9 = 0

x^2-9

 

  Start with:   x2 − 9 = 0
  Move 9 to right:   x2 = 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 = 42 = 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

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 we continued this we 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:

square-root-exponent-half
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.

 

Harder Question