GCSE Maths Algebra Laws of indices

Brackets with indices

Brackets with Indices

Here we will learn about brackets with indices.

There are also laws of indices worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are brackets with indices?

Brackets with indices are where we have a term inside a bracket with an index (or power) outside of the bracket.
To do this we can raise everything inside the bracket to the power. 

E.g.

\[(a^{4})^{2}={a}^{4}\times{a}^{4}={a}\times{a}\times{a}\times{a}\times{a}\times{a}\times{a}\times{a}=a^{8}\]

We could also have used the multiplication law of indices

\[(a^{4})^{2}={a}^{4}\times{a}^{4}={a}^{4+4}=a^{8}\]

However, a quicker method would be to multiply the indices:

\[(a^{4})^{2}={a}^{4\times2}={a}^{8}\]

In general when there is a term inside a bracket with an index (or power) outside of the bracket multiply the powers.

\[(a^{m})^{n}=a^{{m}\times{n}}={a}^{mn}\]

Brackets with indices is one of the laws of indices.

What are brackets with indices?

What are brackets with indices?

How to work out brackets with indices

In order to work out brackets with indices:

  1. Raise the term inside the brackets by the power outside the brackets
  2. Make sure you have considered the coefficient
  3. Write the final answer

How to work out brackets with indices

How to work out brackets with indices

Brackets with indices worksheet

Get your free brackets with indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Brackets with indices worksheet

Get your free brackets with indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Brackets with indices examples

Example 1: single number base

Write as a single power of 5:

\[(5^3)^2\]

  1. Raise the term inside the brackets by the power outside the brackets

\[(5^3)^2=5^3\times 5^3=5^{3+3}=5^6\]

It is quicker to multiply the indices (powers) together.

\[(5^3)^2=5^{3\times2}=5^6\]

2Make sure you have considered the coefficient

There is no coefficient to consider.

3Write the final answer

The question asked for the answer to be as a single power so the final answer is:

\[5^6\]

Example 2: single number base

Write as a single power of 7

\[(7^2)^4\]
\[(7^2)^4=7^2\times 7^2\times 7^2\times 7^2 =7^{2+2+2+2}=7^8\]

It is quicker to multiply the indices (powers) together:

\[(7^2)^4=7^{2\times4}=7^8\]

There is no coefficient to consider.

The question asked for the answer to be as a single power so the final answer is:

\[7^8\]

Example 3: algebraic base with coefficient of 1

Write as a single power:

\[(x^3)^4\]
\[(x^3)^4=x^3\times x^3\times x^3\times x^3 =x^{3+3+3+3}=x^{12}\]

It is quicker to multiply the indices (powers) together:

\[(x^3)^4=x^{3\times4}=x^{12}\]

The coefficient is 1 and does not need special consideration.

The question asked for the answer to be as a single power so the final answer is:

\[x^{12}\]

Example 4: algebraic base with coefficient of 1

Write as a single power:

\[(y^4)^5\]
\[(y^4)^5=y^4\times y^4\times y^4\times y^4 \times y^4=y^{4+4+4+4+4}=y^{20}\]

It is quicker to multiply the indices (powers) together.

\[(y^4)^5=y^{4\times5}=y^{20}\]

The coefficient is 1 and does not need special consideration.

The question asked for the answer to be as a single power so the final answer is:

\[y^{20}\]

Example 5: algebraic base with a coefficient greater than 1

Simplify:

\[(4y^2)^3\]

You can split the term inside the bracket into the coefficient and the base with its index (power).

The base and its index is:

\[y^2\]

This is being raised to the power 3.

It is quicker to multiply the indices (powers) together:

\[(y^2)^3=y^{2\times3}=y^6\]

The coefficient is 4. This also needs raising to the power of 3.

\[4^3 = 4\times4\times4=64\]

Altogether it would be:

\[(4y^2)^3=4^3 \times y^{2\times3}=64y^6\]

The question asked for the answer to be as a single power so the final answer is:

\[64y^6\]

Example 6: algebraic base with a coefficient greater than 1

Simplify:

\[(3a^5)^2\]

You can split the term inside the bracket into the coefficient and the base with its index (power).

The base and its index is:

\[a^5\]

This is being raised to the power 2.

It is quicker to multiply the indices (powers) together:

\[(a^5)^2=a^{5\times2}=a^{10}\]

The coefficient is 3. This also needs raising to the power of 2.

\[3^2 = 3\times3=9\]

Altogether it would be:

\[(3a^5)^2=3^2 \times a^{5\times2}=9a^{10}\]

The question asked for the answer to be as a single power so the final answer is:

\[9a^{10}\]

Common misconceptions

  • When simplifying you do not need the multiplication sign between parts of a term 

You do not need a multiplication sign between the coefficient and the algebraic letter.

\[(5x^3)^2=5^2\times x^{3\times2}=25\times x^6\]

So final answer would be:

\[25x^6\]

  • Make sure that you raise everything inside the brackets to the power outside the bracket

It is a common error to forget to raise the coefficient to the power outside of the fraction.  In the example below, it is easy to forget to square the coefficient 4.

\[(4a^6)^2=4^2\times a^{6\times2}=16\times a^{12}\]

Practice brackets with indices questions

1. Write as a number to a single power: (2^3)^4

2^{12}
GCSE Quiz True

2^{34}
GCSE Quiz False

2^7
GCSE Quiz False

4096
GCSE Quiz False
(2^3)^4=2^{3\times4}=2^{12}

2. Write as a number to a single power: (7^2)^3

7^{23}
GCSE Quiz False

7^5
GCSE Quiz False

7^6
GCSE Quiz True

117 649
GCSE Quiz False
(7^2)^3=7^{2\times3}=7^6

3. Write as a single power: (x^4)^2

x^{42}
GCSE Quiz False

x^8
GCSE Quiz True

x^2
GCSE Quiz False

x^6
GCSE Quiz False
(x^4)^2=x^{4\times2}=x^8

4. Write as a single power: (h^9)^7

h^{97}
GCSE Quiz False

h^2
GCSE Quiz False

h^{16}
GCSE Quiz False

h^{63}
GCSE Quiz True
(h^9)^7=h^{9\times7}=h^{63}

5. Simplify: (2d^3)^2

2d^6
GCSE Quiz False

2d^5
GCSE Quiz False

4d^6
GCSE Quiz True

4d^5
GCSE Quiz False
(2d^3)^2=2^2\times d^{3\times2}=4d^6

6. Simplify: (5e^4)^3

125e^{12}
GCSE Quiz True

5e^{12}
GCSE Quiz False

5e^7
GCSE Quiz False

125e^7
GCSE Quiz False
(5e^4)^3=5^3\times e^{4\times4}=125e^{12}

Brackets with indices GCSE questions

1. Simplify (3p^3 q^4)^2

 

(2 marks)

Show answer
9\times p^6\times q^8

 

for 2 of 3 terms correct

(1)

9p^6 q^8

 

for correct final answer

(1)

2. Simplify (2n^2 m^5)^3

 

(2 marks)

Show answer
8\times n^6\times m^{15}

 

for 2 of 3 terms correct

(1)

8n^6 m^{15}

 

for correct final answer

(1)

3. Simplify (7xy^4)^2

 

(2 marks)

Show answer
49\times x^2\times y^8

 

for 2 of 3 terms correct

(1)

49x^2 y^8

 

for correct final answer

(1)

Learning checklist

You have now learned how to:

  • Simplify brackets with indices 

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