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:

square root of 9 is 3
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) ?

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 if we had squared a positive number:

5x5 = -5x-5

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

Note: if someone says "minus 5 squared" do we:

  • Square the 5, then do the minus?
  • Or do we square (-5) ?

We get different answers:

Square 5, then do the minus:   Square (-5):
-(5×5) = -25   (-5)×(-5) = +25

Always make it clear what you mean, and that is what the "( )" are for.

 

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 I 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 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

We can also square 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?

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:


we would say "square root of 9 equals 3"

Example: What is √25?

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

So the answer is:

√25 = 5

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!

 

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!  

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 would be a good start.