Squaring

Squaring is where you multiply a number by itself.

For example, instead of writing 3 x 3 you can write 3².

The ² is known as an index number or power.

So, 2 x 2 = 2² or 8 x 8 = 8² and so on.

Square numbers are the results of squaring a numbers: 1, 2, 4, 9 and so on.

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Square roots

A square root is the opposite of a square number.

So, the square root of 4 is 2 or the square root of 9 is 3.

The symbol used for a square root is

However, an important point to remember is that if you square a minus number that also makes a positive. So, the square root of 4 isn’t just 2 it’s also -2 because:

-2² = -2 x -2 = 4

All positive numbers have two square roots.

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Cubing

Cubing refers to multiplying a number by itself three times.

1 x 1 x 1 = 1³ = 1

2 x 2 x 2 = 2³ = 8

3 x 3 x 3 = 3³ = 27 and so on.

Similarly with squaring, you have cube numbers: 1, 8, 27 and so on.

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Cube roots

A cube root is the opposite of a cube number.

The symbol used for a cube root is

So, to work backwards from cube numbers, the cube root of 8 is 2 and the cube root of 27 is 3.

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Rules of indices

It is possible to do calculations on squared or cubed numbers as well as numbers raised to the power of another number.

All you need to do is remember the following rules.

For multiplying you add all the powers together:

3³ x 3⁷ = (3 x 3 x 3) x (3 x 3 x 3 x 3 x 3 x 3 x 3) = 3¹⁰

If you want to divide then you subtract the powers:

Lastly, if you want to take the power of a number already raised to a power then you multiply the powers together:

(4²)³ = (4 x 4) x (4 x 4) x (4 x 4) = 4⁶

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Powers and roots (Higher Tier)

For the higher tier you’ll need to know about standard index form and also another range of fractional powers.

Standard index form

The mass of the earth is a big number. To write it out fully would be 597 followed by 22 zeros! In caluculations, this would take a long time, so we can simplify things by using standard form. Standard form is an easier way to write very big or very small numbers.

The layout of a standard index form number is always:

A x 10ⁿ

‘A’ will only ever be between 1 and 10.

For example, say you wanted to write 24 000 000 in standard index form.

You can rewrite this into:

2.4 x 10 000 000

Which can also be shown as:

2. 4 x 10 x 10 x 10 x 10 x 10 x 10 x 10

Which, in standard index form, is written as:

2.4 x 10⁷

You can also work the other way. For example, say you hard to write 5 x 10⁵ as an ordinary number.

5 x 10⁵ = 5 x (10 x 10 x 10 x 10 x 10) = 500 000

A good rule of thumb is that the power tells you how many 0s you need.

In order to subtract or add numbers in standard index form you first need to convert them back into ordinary numbers.

For instance: 7 x 10⁵ – 5.6 x 10⁴

In ordinary numbers this would be:

700 000 – 560 000 = 140 000

When you multiply or divide you can use the same rules as with multiplying and dividing powers.

So, for example: (4 x 10³) x (5 x 10⁶)

Multiply the numbers together and the powers together:

(4 x 10³) x (5 x 10⁶) = 20 x 10¹⁸

Powers are also possible as zero, negative numbers and fractions, each of which has their own rules.

Any number to the power of zero always equals 1:

a = 1

A number to the power of a negative number (a^-b) can also be written as:

For example:

A number to the power of