Squares And Square Roots In Algebra - Math Is Fun

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 r whose square is x:

r2 = x

The Square Root Symbol

This is the special symbol that means "square root", it is like a check mark, 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:

we say "square root of 9 equals 3"

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Note: in simple cases we can write √9 but otherwise use an overline √ab or parentheses √(ab) to show what is included.

Negative Numbers

Negative numbers can also be squared.

Example: What is minus 5 squared?

...

... but ... 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 "( )".

Corrected 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.

Exactly the same as when we square a positive number:

Now remember our definition of a square root?

A square root of x is a number r whose square is x:

r2 = x

And we just found that:

(+5)2 = 25 (−5)2 = 25

There are two numbers whose square makes 25

So both +5 and −5 are square roots of 25

Two Square Roots

There can be a positive and negative square root!

This is important to remember.

Example: Solve w2 = a

Answer:

w = √a and w = −√a

Principal Square Root

If there are two square roots, why do we say √25 = 5 ?

The symbol √ means the principal (non-negative) square root

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 (but it can be zero).

Plus-Minus Sign

±	is a special symbol that means "plus or minus",

so instead of writing:	w = √a and 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

Start with: x2 − 9 = 0 Move 9 to right: x2 = 9 Square Roots: 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 Square Roots: x − 3 = ±√16 Calculate √16: x − 3 = ±4 Add 3 to both sides: 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

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 that leads to a 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

Why does √xy = √x√y ?

We can use the fact that squaring a square root gives us the original value back again:

(√a)2 = a

Assuming a is not negative!

We can do that for xy:(√xy)2 = xy And also to x, and y, separately:(√xy)2 = (√x)2(√y)2 Use a2b2 = (ab)2:(√xy)2 = (√x√y)2 Remove square from both sides:√xy = √x√y

An Exponent of a Half

A square root can also be written as a fractional exponent of one-half:

√x = x½ 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.

457, 458, 1084, 1085, 1086, 2286, 2287, 3994, 3995, 3996 Irrational Numbers Surds Scientific Calculator Algebra Index