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 = powers square = 3 × 3 = 9

 

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

4 squared is 16
This says "4 Squared equals 16"
(the little 2 says the number appears twice in multiplying)

Squares From 02 to 62

0 Squared = 02 = 0 × 0 = 0
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:
  squares in times table

Negative Numbers

We can also square negative numbers.

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

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 squaring a positive number:

5x5 = -5x-5

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

Square Roots

A square root goes the other way:

square root of 9 is 3

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.

It is like asking:

What can we multiply by itself to get this?

tree root

To help you remember think of the root of a tree:

"I know the tree, but what root made it?"

In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

square vs square root
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 discovered earlier 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

radical 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:

square root of 9 is 3
and we 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.

So the answer is:

√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 (or zero) result.

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The Perfect Squares (also called "Square Numbers") are the squares of the integers:

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc
Perfect Squares: 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...

Try to remember them up to 12.

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!

calculator

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

square root button   Use your calculator's square root button!   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)
around 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 is a good start.

Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like 4/4/16

The next after that is the 5th of May 2025 (5/5/25)