Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

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 says the number appears twice in multiplying)

Squares From 12 to 62

 1 Squared = 12 = 1 × 1 = 1 2 Squared = 22 = 2 × 2 = 4 3 Squared = 32 = 3 × 3 = 9 4 Squared = 42 = 4 × 4 = 16 5 Squared = 52 = 5 × 5 = 25 6 Squared = 62 = 6 × 6 = 36

 The squares are also on the Multiplication Table:

Negative Numbers

We can also square negative numbers.

Example: What happens when we square (−5) ?

(−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 if we had squared a positive number:

(For more detail read Squares and Square Roots in Algebra)

Square Roots

A square root goes the other way:

3 squared is 9, so a square root of 9 is 3

A square root of a number is ...

... a value that can be multiplied by itself to give the original number.

A square root of 9 is ...

... 3, because when 3 is multiplied by itself we get 9.

What can we multiply by itself to get this?

 To help you remember think of the root of a tree: "I know the tree, but what is the root that produced it?" In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

 4 16 5 25 6 36

Decimal Numbers

It also works for decimal numbers.

Try the sliders below. Note: the numbers here are only shown to 2 decimal places.

Using the sliders (remembering it is only accurate to 2 decimal places):

• What is the square root of 8?
• What is the square root of 9?
• What is the square root of 10?
• What is 1 squared?
• What is 1.1 squared?
• What is 2.6 squared?

Negatives

We found out before that we can square negative numbers:

Example: (−3) squared

(−3) × (−3) = 9

And of course 3 × 3 = 9 also.

So the square root of 9 could be −3 or +3

Example: What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

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

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 mathematics look important!

We use it like this:

we would say "square root of 9 equals 3"

Example: What is √25?

Well, we just happen to know that 25 = 5 × 5, so when we multiply 5 by itself (5 × 5) we will get 25.

√25 = 5

But wait a minute! Can't the square root also be −5? Because (−5) × (−5) = 25 too.

• Well the square root of 25 could be −5 or +5.
• But when we use the radical symbol we only give the positive result.

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The perfect squares are the squares of the whole numbers:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc Perfect Squares: 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...

Try to remember at least the first 10 of those.

Calculating Square Roots

It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

Example: what is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

• Let's try 3.5: 3.5 × 3.5 = 12.25
• Let's try 3.2: 3.2 × 3.2 = 10.24
• Let's try 3.1: 3.1 × 3.1 = 9.61
• ...

Getting closer to 10, but it will take a long time to get a good answer!

 At this point, I get out my calculator and it says: 3.1622776601683793319988935444327 But the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation !

Note: numbers like that are called Irrational Numbers, if you want to know more.

The Easiest Way to Calculate a Square Root

 Use your calculator's square root button!

And also use your common sense to make sure you have the right answer.

A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

 a) start with a guess (let's guess 4 is the square root of 10) b) divide by the guess (10/4 = 2.5) c) add that to the guess (4 + 2.5 = 6.5) d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25) e) now, set that as the new guess, and start at b) again

• Our first attempt got us from 4 to 3.25
• Going again (b to e) gets us: 3.163
• Going again (b to e) gets us: 3.1623

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

3.1623 x 3.1623 = 10.00014

Now ... why don't you try calculating the square root of 2 this way?

How to Guess

What if we have to guess the square root for a difficult number such as "82,163" ... ?

In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 would be a good start.