Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

Example: What's 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 5²

The squares are also on the Multiplication Table:

Negative Numbers

What happens when we square a negative number?

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 squares and square roots:

1		1
2		4
3		9
4		16
5		25
6		36
7		49

Generally

It also works generally.

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

Use the sliders to answer these questions:

• What's the square root of 8?
• What's the square root of 9?
• What's the square root of 10?
• What's 1 squared?
• What's 1.1 squared?
• What's 0.5 squared?
• What's 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 check mark,
and it likely evolved from the letter "r" for radix (Latin for root).

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's 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 × 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 (100×100=10,000), and the square root of 8 (the first digit) is about 3 (3×3=9), so 300 is a good start.