# Simplify Terms with Brackets

In this worksheet, students will expand and simplify brackets, applying index laws when working with powers.

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

You may already know how to expand simple bracket such as:

5(b + 3) = 5b + 15

5a(b + 4) = 5ab + 20a

Now we are going to look at expanding brackets where some of the terms involve indices.

Remember that indices is just another way of saying powers

We can use index laws to help us simplify terms when expanding brackets.

Remember that when we multiply out numbers with indices, we can just add the powers:

e.g. w³ × w² = w5

Let's apply this to expanding brackets.

e.g. Expand 5a²(a4 + 2a)

5a² × a4 = 5a6

5a² × 2a = 10a³

So the answer is: 5a6 + 10a³

In this activity, we will expand brackets to create simpler expressions, applying index laws to calculate successfully when working with powers.

Expand:

3a3(a5 - 3a)

3a15 - 9a3

3a8 - 9a4

3a8 + 9a4

3a8 - 3a4

Match each expression below with brackets to its correctly expanded version.

## Column B

x(x2 + 2)
15x5 + 6x3
3x(x2 + 2)
3x5 + 6x
3x3(x2 + 2)
3x3 + 6x
3x3(5x2 + 2)
x3 + 2x

Expand this bracket:

2y3(2y4 - 3y)

2y7 - 2y4

2y7 - 3y4

4y12 - 6y3

4y7 - 6y4

Which of the options below can be expanded to give 36x8 - 12x5?

6x5(6x3 - 2)

6x5(6x3 - 2)

4x2(9x6 - 3x3)

9x4(4x2 - 3x)

Your friend has expanded 7x3(3x² + 2x) to reach the answer below.

But they have made a mistake!

Which of their terms below is incorrect?

21x6 + 14x4

Expand:

5x2(2x2 + 3x4 - x3)

10x4 + 15x8 - 5x6

10x4 + 15x6 - 5x5

7x4 + 8x6 - 4x5

7x4 + 8x8 - 4x6

Match each bracketed expression below to its correct expansion.

## Column B

2x2(3x3 + 4x4 + 5...
6x5 + 8x6 + 10x7
3x2(3x3 + 4x4 + 5...
8x5 + 10x6 + 12x7
2x2(4x3 + 5x4 + 6...
12x5 + 15x6 + 18x7
3x2(4x3 + 5x4 + 6...
9x5 + 12x6 + 15x7

Expand:

4x(5x2 - 2xy)

9x3 - 6xy

20x3 - 8x2

20x3 - 8x2y

9x3 - 6x2y

Only two of the sets of brackets below have been expanded correctly.

Which two are correct?

2gh(3g + 5g²) = 6g²h + 10g³h

2gh(g + 5g²) = 2gh + 10g³

2gh(3gh + 5g²) = 6g²h + 10g³h

2gh(3gh + 5g²) = 6g²h² + 10g³h

Expand:

5a³b²(2ab² + 4a²b²)

7a4b4 + 9a5b4

7a4b4 + 9a6b4

10a3b4 + 20a6b4

10a4b4 + 20a5b4

• Question 1

Expand:

3a3(a5 - 3a)

3a8 - 9a4
EDDIE SAYS
We need to calculate each term separately: 3a³ × a5 = 3a8 3a³ × -3a = - 9a4 If we put these terms back together, we reach the correct answer of: 3a8 - 9a4 How did you find that?
• Question 2

Match each expression below with brackets to its correctly expanded version.

## Column B

x(x2 + 2)
x3 + 2x
3x(x2 + 2)
3x3 + 6x
3x3(x2 + 2)
3x5 + 6x
3x3(5x2 + 2...
15x5 + 6x3
EDDIE SAYS
Make sure you use index laws when expanding brackets. Let's have a look at: 3x3(5x² + 2) 3x3 × 5x² = 15x5 3x3 × 2 = 6x3 If we put this back together, we reach: 3x3(5x2 + 2) Can you work out the other matches independently, using the example above to help you? Remember that when no power is present, it is the same as x1.
• Question 3

Expand this bracket:

2y3(2y4 - 3y)

4y7 - 6y4
EDDIE SAYS
You need to use index laws to expand this bracket. 2y³ × 2y4 = 4y7 2y³ × - 3y = - 6y4 So our answer is: 4y7 - 6y4
• Question 4

Which of the options below can be expanded to give 36x8 - 12x5?

6x5(6x3 - 2)
6x5(6x3 - 2)
4x2(9x6 - 3x3)
EDDIE SAYS
The only bracket that will not expand to 36x8 - 12x5 is: 9x4(4x2 - 3x) Let's see why... 9x4 × 4x2 = 36x6 9x4 × -3x) = -27x5 So, overall, it expands to: 36x6 - 27x5
• Question 5

Your friend has expanded 7x3(3x² + 2x) to reach the answer below.

But they have made a mistake!

Which of their terms below is incorrect?

21x6 + 14x4
EDDIE SAYS
21x6 is incorrect. 7x3 × 3x² = 21x5 not 21x6 We must remember to add the powers, not multiply them, when we multiplying terms.
• Question 6

Expand:

5x2(2x2 + 3x4 - x3)

10x4 + 15x6 - 5x5
EDDIE SAYS
This may look tricky, but we just need to take it one step at a time. 5x2 × 2x2 = 10x4 5x2 × 3x4 = 15x6 5x2 × -x³ = -5x5 If we put these elements back together again, the correct answer is: 10x4 + 15x6 - 5x5
• Question 7

Match each bracketed expression below to its correct expansion.

## Column B

2x2(3x3 + 4...
6x5 + 8x6 +...
3x2(3x3 + 4...
9x5 + 12x6 ...
2x2(4x3 + 5...
8x5 + 10x6 ...
3x2(4x3 + 5...
12x5 + 15x6...
EDDIE SAYS
In this question, you can choose to either expand the sets of brackets on the left, or factorise the expressions on the right. Remember that these are opposite functions. Let's use one example to show how we can use either method. 1) Expand: 2x2 × 3x3 = 6x5 2x2 × 4x4 = 8x6 2x2 × 5x5 = 10x7 Altogether this gives: 6x5 + 8x6 + 10x7 2) Factorise: All terms in 6x5 + 8x6 + 10x7 have a common factor of 2x2. So we put this outside our bracket like this: 2x2( __ _ __ _ __ ) To find what goes inside the brackets, we need to divide each term by 2x2. 6x5 ÷ 2x2 = 3x3 8x6 ÷ 2x2 = 4x4 10x7 ÷ 2x2 = 5x5 Both these answers match, whichever way we choose to approach them. Find the remaining matches using your preferred method, and this example, to support you.
• Question 8

Expand:

4x(5x2 - 2xy)

20x3 - 8x2y
EDDIE SAYS
Let's multiply everything inside the brackets, by the multiplier on the outside: 4x × 5x² = 20x³ 4x × -2xy = - 8x²y Don't forget about the y here. You cannot simplify it using index law, but you must not let it disappear either!
• Question 9

Only two of the sets of brackets below have been expanded correctly.

Which two are correct?

2gh(3g + 5g²) = 6g²h + 10g³h
2gh(3gh + 5g²) = 6g²h² + 10g³h
EDDIE SAYS
Did you remember to apply the laws of indices to each of the terms? 2gh(g + 5g²) = 2gh + 10g³ is incorrect. If we multiply both terms by 2gh, we reach 2g²h + 10g³h, not 2gh + 10g³h. 2gh(3gh + 5g²) = 6g²h + 10g³h is also incorrect. The correct answer should be 6g²h² + 10g³h. The other two options have been expanded correctly.
• Question 10

Expand:

5a³b²(2ab² + 4a²b²)

10a4b4 + 20a5b4
EDDIE SAYS
Wow, this one looks confusing! Let's just take our time and multiply each term by the number outside the brackets: 5a³b² × 2ab² = 10a4b4 5a³b² × 4a²b² = 20a5b4 If we put this back together, we reach: 10a4b4 + 20a5b4 That wasn't so tricky, was it? Stellar job completing this activity! Hopefully those index laws are becoming second nature to you now.
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