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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.

**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**.

In order to work out brackets with indices:

**Raise the term inside the brackets by the power outside the brackets****Make sure you have considered the coefficient****Write the final answer**

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

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

COMING SOONBrackets with indices is part of the larger topic, laws of indices. It may be useful to explore the main topic before looking into the detailed individual lessons below:

Write as a single power of

\[(5^3)^2\]

**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.

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

2**Make sure you have considered the coefficient**

There is no coefficient to consider.

3**Write the final answer**

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

\[5^6\]

Write as a single power of

\[(7^2)^4\]

**Raise the term inside the brackets by the power outside the brackets**

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

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

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

**Make sure you have considered the coefficient**

There is no coefficient to consider.

**Write the final answer**

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

\[7^8\]

Write as a single power:

\[(x^3)^4\]

**Raise the term inside the brackets by the power outside the brackets**

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

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

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

**Make sure you have considered the coefficient**

The coefficient is

**Write the final answer**

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

\[x^{12}\]

Write as a single power:

\[(y^4)^5\]

**Raise the term inside the brackets by the power outside the brackets**

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

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

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

**Make sure you have considered the coefficient**

The coefficient is

**Write the final answer**

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

\[y^{20}\]

Simplify:

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

**Raise the term inside the brackets by the power outside the brackets**

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

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

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

**Make sure you have considered the coefficient**

The coefficient is

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

Altogether it would be:

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

**Write the final answer**

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

\[64y^6\]

Simplify:

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

**Raise the term inside the brackets by the power outside the brackets**

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

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

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

**Make sure you have considered the coefficient**

The coefficient is

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

Altogether it would be:

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

**Write the final answer**

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

\[9a^{10}\]

**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

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

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

2^{12}

2^{34}

2^7

4096

(2^3)^4=2^{3\times4}=2^{12}

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

7^{23}

7^5

7^6

117 649

(7^2)^3=7^{2\times3}=7^6

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

x^{42}

x^8

x^2

x^6

(x^4)^2=x^{4\times2}=x^8

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

h^{97}

h^2

h^{16}

h^{63}

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

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

2d^6

2d^5

4d^6

4d^5

(2d^3)^2=2^2\times d^{3\times2}=4d^6

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

125e^{12}

5e^{12}

5e^7

125e^7

(5e^4)^3=5^3\times e^{4\times4}=125e^{12}

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)**

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)**

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)**

for correct final answer

**(1)**

You have now learned how to:

- Simplify brackets with indices

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