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Here we will learn about powers, raising numbers to the power of, and writing expressions in index form.

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

**To the power of** is used to describe a number raised to a power.

For example,

- We can write 3 \times 3 in a shorter way, using exponents 3^{2}.

We can say this as 3**to the power of**2.

- We can write -2 \times -2 \times -2 in a shorter way, using exponents -2^{3}.

We can say this as -2**to the power of**3.

- We can write \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} in a shorter way, using exponents \frac{1}{5}^{4}.

We can say this as \frac{1}{5} to the power of 4.

This is known as** index notation** or **index form** and features heavily in scientific notation. To write a number in index form we have a **base number** raised to a **power**. The base number can be an integer, decimal, fraction, etc. and the power tells us the number of times we multiply the base number.

For example,

The number 6 is called the base, and the number 2 is the exponent (or power).

In words, 6^2 can be written as “ 6 **to the power of** 2 ” or “ 6 to the second power”, or “ 6 squared”.

We can multiply a number by itself as many times as we want using powers.

In order to describe powers with words:

**Identify the base number.****Consider the index (or power).****Work out the answer.**

Get your free to the power of worksheet of 20+ powers and roots questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free to the power of worksheet of 20+ powers and roots questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**To the power of** is part of our series of lessons to support revision on **powers and roots**. You may find it helpful to start with the main powers and roots 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:

Describe the expression 11 \times 11 \times 11 \times 11.

**Identify the base number.**

The base number is 11.

2**Consider the index (or power).**

11 is being multiplied by itself 4 times. So the power will be 4.

3**Work out the answer.**

11 to the power of 4, or 11 to the fourth power.

Describe the expression 7^{3}.

**Identify the base number.**

The base number is 7.

**Consider the index (or power).**

7^3=7 \times 7 \times 7

The index (or power) is 3. \ 7 is being multiplied by itself 3 times.

**Work out the answer.**

7 to the power of 3 or 3 to the third power.

Evaluate 2^{4}.

**Identify the base number.**

The base number is 2.

**Consider the index (or power).**

The index (or power) is 4.

**Work out the answer.**

2^4=2 \times 2 \times 2 \times 2=16

Evaluate 3^{5}.

**Identify the base number.**

The base number is 3.

**Consider the index (or power).**

The index (or power) is 5.

**Work out the answer.**

3^5=3 \times 3 \times 3 \times 3 \times 3=243

Write 9 \times 9 \times 9 \times 9 \times 9 in index form.

**Identify the base number.**

There is one base number, 9.

**Consider the index (or power).**

9 is multiplied by itself 5 times, the index is 5.

**Work out the answer.**

9 \times 9 \times 9 \times 9 \times 9=9^5

Write 5 \times 3 \times 5 \times 3 \times 5 in index form.

**Identify the base number.**

There are two base numbers, 3 and 5.

**Consider the index (or power).**

3 is multiplied by itself 2 times, its index is 2.

5 is multiplied by itself 3 times, its index is 3.

**Work out the answer.**

5\times 3 \times 5 \times 3 \times 5=3^2 \times 5^3

**The power is what we multiply the number by**

The power tells us how many times to multiply the base number by itself.

For example,

4^2=4 \times 4=16 not 4 \times 2=8.

**Mixed bases in index form**

Mixed bases in multiplication expressions require a power per base number when written in index form.

For example,

5 \times 0.5 \times 5 \times 5 \times 0.5 \times 5=0.5^2 \times 5^4

**Power of**\bf{1}

A key fact to remember is that every number has a secret power of 1, we can’t see it, but we have to remember it’s there.

For example,

\begin{aligned} &8=8^1 \\\\ &80=80^1 \\\\ &800=800^1 \\\\ &\frac{1}{8}=\left(\frac{1}{8}\right)^1 \\\\ &1.8=1.8^1 \end{aligned}

1. Evaluate 5^{2}.

10

25

5 \times 5

5 to the power of 2

5^2=5 \times 5=25

2. Write 2 \times 7 \times 7 \times 2 \times 7 \times 7 in index form.

2^2 \times 7^4

2 \times 7^2

2^4 \times 7

(2 \times 7)^6

There are two base numbers, 2 and 7.

2 is multiplied by itself 2 times, it has a power of 2.

7 is multiplied by itself 4 times, it has a power of 4.

2 \times 7 \times 7 \times 2 \times 7 \times 7=2^2 \times 7^4

3. Evaluate 4^{4}.

16

4 \times 4 \times 4 \times 4

256

4 to the power of 4

4^4=4 \times 4 \times 4 \times 4=256

4. Describe the expression 5^{7}.

5 to the power of 7

5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5

7 to the power of 5

35

The base number is 5.

The power is 7.

Hence, 5 to the power of 7.

5. Write 0.3 \times 3 \times 0.3 \times 0.3 \times 0.3 in index form.

3^4 \times 0.3

(0.3 \times 3)^5

0.3^4 \times 3

0.3 \times 3^4

There are two base numbers, 0.3 and 3.

0.3 is multiplied by itself 4 times, it has a power of 4.

3 is multiplied by itself 1 times, it has a power of 1.

0.3 \times 3 \times 0.3 \times 0.3 \times 0.3=0.3^4 \times 3

6. Describe the expression 7^{3}.

7 \times 7 \times 7

343

7 to the power of 3

3 to the power of 7

The base number is 7.

The power is 3.

Hence, 7 to the power of 3.

1. Evaluate 8^4.

**(1 mark)**

Show answer

4096

**(1)**

2. Write 4 \times 6 \times 6 \times 4 \times 4 \times 5 in index form.

**(1 mark)**

Show answer

4^3\times 5 \times 6^2

**(1)**

3. Simplify \frac{3 \times 3\times 3\times 3}{4\times 4}.

**(2 marks)**

Show answer

Evaluating the numerator 3\times 3 \times 3 \times 3=3^{4}=81

or evaluating the denominator 4 \times 4 =4^{2}=16.

**(1)**

Writing the resulting quotient \frac{3^{4}}{4^{2}}.

**(1)**

You have now learned how to:

- Describe numbers to the power of
- Identify special powers

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