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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.

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

DOWNLOAD FREE**Brackets with indices** is part of our series of lessons to support revision on **laws of indices**. You may find it helpful to start with the main laws of indices lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

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

- Proof maths
- Functions in algebra
- Sequences

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