 # Divide Terms Involving Index Laws

In this worksheet, students will divide terms using index laws to simplify expressions. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Algebraic Expressions

Difficulty level:   ### QUESTION 1 of 10

Let us first recap the third index law:

am ÷ an = am-n

We can use this to simplify algebraic expressions in the same way as we do when working with numbers.

This may look tricky, but it's much simpler in practise!

When dividing two (or more) numbers or variables which have the same base (big number), we can use the third index law to simplify the expressions by simply subtracting the powers.

e.g. x12 ÷ x6 = x6

If you have two or more numbers, we need to remember to divide them as well.

e.g. 16b8 ÷ 8b5 = 2b3

In this activity, we will use index laws (like the example above) to simplify algebraic expressions to make them easier to work with.

Match each expression below to its simplified form.

## Column B

a7 ÷ a2
a2
a5 ÷ a2
a5
a9 ÷ a3
a6
a7 ÷ a2 ÷ a
a3

Simplify:

y12 ÷ y3

y5

y9

15

19

Simplify:

27x6 ÷ 3x3

9x2

24x2

9x3

24x3

Which one of these expressions can be simplified using the laws of indices?

a3 ÷ a2

a6 ÷ b6

x7 ÷ y5

x5 ÷ x7

Use the laws of indices to decide which of the statements below are true.

Which of the expressions below could be simplified to 4x2?

16x4 ÷ 4x2

24x10 ÷ 6x8

24x8 ÷ 6x2

3x10 ÷ 2x7

Match each expression below to their simplified version.

## Column B

15x8 ÷ 3x4
4x
24x5 ÷ 6x4
10x-2
12x9 × 2x3
5x4
33x ÷ 3x3
6x6

Which of the expressions below will simplify to give 3x2y3?

3x2 ÷ 12x4

9x4y ÷ 2x4y5

45x6y7 ÷ 15x4y4

6x88 ÷ 6y6

Simplify:

10a7b6c5 ÷ 2abc

8a6b5c4

5a6b5c4

5a7b6c5

8a7b6c5

Simplify:

20w3x5 ÷ 5wx3 ÷ 2wx

13w3x

13wx

2w3x

2wx

• Question 1

Match each expression below to its simplified form.

## Column B

a7 ÷ a2
a5
a5 ÷ a2
a3
a9 ÷ a3
a6
a7 ÷ a2
a2
EDDIE SAYS
Remember that when we divide powers of the same whole number, we can just subtract the powers. So a7 ÷ a2 = a7 - 2 = a5 It's as simple as that!
• Question 2

Simplify:

y12 ÷ y3

y9
EDDIE SAYS
Both terms have y as a base, so we can just subtract the powers to reach the answer: 12 − 3 = 9 Remember to keep the base number the same though! The correct answer is: y9
• Question 3

Simplify:

27x6 ÷ 3x3

9x3
EDDIE SAYS
Remember to divide the numbers first: 27 ÷ 3 = 9 Then apply the laws of indices to x6 ÷ x3 = x6 - 3 = x3 So our final answer is: 9x3
• Question 4

Which one of these expressions can be simplified using the laws of indices?

a3 ÷ a2
x5 ÷ x7
EDDIE SAYS
The laws of indices can only be used if we have the same base number or variable. Options 2 and 3 have different variables (a and b; x and y), so we cannot simplify them using the laws of indices. Does that make sense?
• Question 5

Use the laws of indices to decide which of the statements below are true.

EDDIE SAYS
Did you spot the false statements? x5 ÷ x7 = x5 - 7 5 - 7 = -2, so this expression should have a negative 2 as its power. p6 ÷ p3 = p6 - 3 6 - 3 = 3 but this option has been written with a square instead (2) which makes it incorrect.
• Question 6

Which of the expressions below could be simplified to 4x2?

16x4 ÷ 4x2
24x10 ÷ 6x8
EDDIE SAYS
Only the first two expressions can be simplified to give 4x2. Remember to divide the whole numbers (or variables) first, and then apply the index laws. In the third option, the numbers are correct (4) but the powers subtract to give 6 not 2. In the fourth option, both the numbers (3/2 or 1.5) and the powers (3) are incorrect.
• Question 7

Match each expression below to their simplified version.

## Column B

15x8 ÷ 3x4...
5x4
24x5 ÷ 6x4...
4x
12x9 × 2x3<...
6x6
33x ÷ 3x3
10x-2
EDDIE SAYS
To find the matching answers, divide the whole numbers (or variables) first and then use index laws to work out what power x needs to take. Remember, if there is no power next to the letter, the value of the power is 1. Let's look at one together. 33x ÷ 3x3 = 33 ÷ 3 = 10 x1 - 3 = x-2 So our complete answer here is: 10x-2 Can you complete the others independently, following this example?
• Question 8

Which of the expressions below will simplify to give 3x2y3?

9x4y ÷ 2x4y5
45x6y7 ÷ 15x4y4
EDDIE SAYS
The whole numbers (or variables) need to divide to give 3, which rules out the first option immediately. Only two of the remaining options are correct. Have you checked the indices? When you subtract the powers of x, they need to make 2. The powers of y should make 3 when you subtract them.
• Question 9

Simplify:

10a7b6c5 ÷ 2abc

5a6b5c4
EDDIE SAYS
Let's break this equestion down: 10 ÷ 2 = 5 a7 ÷ a = a6 b6 ÷ b = b5 c5÷ c = c4 So our final answer is: 5a6b5c4 That wasn't so difficult once we broke it down, was it?
• Question 10

Simplify:

20w3x5 ÷ 5wx3 ÷ 2wx

2wx
EDDIE SAYS
There are three parts to divide here, but we still need to follow the same rules. 20 ÷ 5 ÷ 2 = 2 w3 ÷ w ÷ w = w x5 ÷ x3 ÷ x = x So our final answer is simply: 2wx Congratulation on completing this activity! If you found you were a little rusty on your index laws, why not try another activity to revise them further?
---- OR ----

Sign up for a £1 trial so you can track and measure your child's progress on this activity.

### What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started 