# Division law for indices

## Home learning focus

In this lesson you will learn how to divide indices.

This lesson includes:

- a video
- a worksheet with answers

# Learn

## Index form

The notation **3²** (three squared) and **2³** (two cubed) is known as **index form**. The small digit is called the **index number** or **power**.

**3²** = 3 × 3 = **9**

**2³** = 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 **2³** you will see that some of the **2s** cancel.

There are **5 twos** from **2⁵** and **3 twos** from **2³**, 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.

Finished? Click here for the answer sheet.

# There's more to learn

Have a look at these other resources around the BBC and the web.