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!

You can use it like this:

square root of 9
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

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

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:

5x5 = -5x-5

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

x^2-9

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:

√(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 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:

square-root-exponent-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.

 

Harder Question