# To The Power Of

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.

## What is meant by to the power of?

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.

## How to describe powers

In order to describe powers with words:

1. Identify the base number.
2. Consider the index (or power).

## To the power of examples

### Example 1: describing powers

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

1. Identify the base number.

The base number is 11.

2Consider the index (or power).

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

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

### Example 2: describing powers

Describe the expression 7^{3}.

The base number is 7.

7^3=7 \times 7 \times 7

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

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

### Example 3: evaluating powers

Evaluate 2^{4}.

The base number is 2.

The index (or power) is 4.

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

### Example 4: evaluating powers

Evaluate 3^{5}.

The base number is 3.

The index (or power) is 5.

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

### Example 5: expressing in index form

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

There is one base number, 9.

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

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

### Example 6: expressing in index form

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

There are two base numbers, 3 and 5.

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

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

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

### Common misconceptions

• 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}

### Practice to the power of questions

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.

### The power of GCSE questions

1. Evaluate 8^4.

(1 mark)

4096

(1)

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

(1 mark)

4^3\times 5 \times 6^2

(1)

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

(2 marks)

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)

## Learning checklist

You have now learned how to:

•  Describe numbers to the power of
•  Identify special powers

## Still stuck?

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#### FREE GCSE maths practice papers (Edexcel, AQA & OCR)

8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.

Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.