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

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:

Definition

Here is the definition:

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"

Negative Numbers

You can also square negative numbers.

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!

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!

The Principal Square Root is sometimes called the Positive Square Root.

Plus-Minus Sign

In a Nutshell

Why Is This 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

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:

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.