Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

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 0² to 6²

The squares are also on the Multiplication Table:

Negative Numbers

We can also square negative numbers.

That was interesting!

When we square a negative number we get a positive result.

Just the same as squaring 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

It is like asking:

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:

4		16
5		25
6		36
7		49

Decimal Numbers

It also works for decimal numbers.

Try the sliders below (note: '...' means the decimals continue on forever):

Use the sliders to answer these questions:

• 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 0.5 squared?
• What is 0.1 squared?

Negatives

We discovered earlier that we can square negative numbers:

And of course 3 × 3 = 9 also.

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

The Square Root Symbol

This is the special symbol that means "square root", it is 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 can use it like this:

and we say "square root of 9 equals 3"

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 with the radical symbol √ we only give the positive (or zero) result.

Perfect Squares

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

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.

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 is a good start.