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 |
| You can also find the squares on the Multiplication Table: |  |
Negative Numbers
You can also square negative numbers.
Example: What happens when you square (-5) ?
Answer:
(-5) × (-5) = 25
(because a negative times a negative gives a positive)
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:
(For more detail read Squares and Square Roots in Algebra)
Note: if someone says "minus 5 squared" do you:
- Square the 5, then do the minus?
- Or do you square (-5) ?
You 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:
3 squared is 9, so a square root of 9 is 3
A square root of a number is ...
A square root of 9 is ...
It is like asking:
What can I 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:
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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
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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) | |
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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.