Squares and Square Roots
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 |
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:

We look at that in more detail in 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 you get 9.
It is like asking:
What can I multiply by itself to get this?
 |
Note: When you see "root" think
"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
You 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
 |
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"
Example: What is √25?
Well, we just happen to know that 25 = 5 × 5, so if you multiply
5 by itself (5 × 5) you 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!
 |
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 you have to guess the square root for a difficult number such as "82,163" ... ?
In that case I would think to myself "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.
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