# Division law for indices

## Home learning focus

In this lesson you will learn how to divide indices.

This lesson includes:

• a video

# Learn

## Index form

The notation (three squared) and (two cubed) is known as index form. The small digit is called the index number or power.

= 3 × 3 = 9

= 2 × 2 × 2 = 8

The index number tells you how many times the number should be multiplied.

## Index laws for division

### Dividing numbers with powers

If you divide 2⁵ by you will see that some of the 2s cancel.

There are 5 twos from 2⁵ and 3 twos from , so after dividing there are 2 twos remaining. The index 2 can be found by subtracting the indices, 5 - 3 = 2.

So 2⁵ ÷ 2³ = 2².

In general, the rule is: 2ᵐ ÷ 2ⁿ = 2⁽ᵐ⁻ⁿ⁾.

Remember, this only works when you are dividing powers of the same base number.

### Examples

2⁵ ÷ 2² = 2⁽⁵⁻²⁾ = 2³

5¹⁰ ÷ 5³ = 5⁽¹⁰⁻³⁾ = 5⁷

45⁹ ÷ 45⁴ = 45⁽⁹⁻⁴⁾ = 45⁵

a⁸ ÷ a² = a⁶

30a²b ÷ 5ab² = 6ab⁻¹

For a short further explanation to aid your understanding of dividing powers with the same base, watch the section between 2:07 and 2:43 in this video from BBC Bitesize, National 5 Maths.

# Practise

## Activity 1

Dividing powers with the same base

Now test your understanding and knowledge of dividing powers with the same base by working through these worksheets from Beyond. Questions are grouped by difficulty either as Bronze, Silver or Gold so aim to get as far as you can! There is also a challenge to complete at the end. Write your answers on a piece of paper.