GCSE Maths Number

Powers and Roots

Cube Numbers and Cube Roots

# Cube Numbers and Cube Roots

Here we will learn about cube numbers and cube roots including what a cube number is, what a cube root is, as well as how to cube a number and how to find the cube root of an integer. You’ll also learn how to solve problems by applying knowledge of cube numbers.

There are also cube numbers and cube roots 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 a cube number?

A cube number is found when we multiply an integer (whole number) by itself three times, these are sometimes called ‘perfect cubes’.
We can cube numbers with decimals places but we do not refer to these as cube numbers.
A number/variable that iscubed is multiplied by itself three times.

E.g.
4 × 4 × 4 can be written as 43 and is spoken as “4 cubed or “4 to the power of 3”

A cube number is found when we multiply an integer (whole number) by itself,

Here are the first 10 cube numbers:

A cube number can be represented as an array which forms the shape of a cube that has a length 3 units, width 3 units and depth 3 units.

E.g.

If we look at 33, this is 3 × 3 × 3

Any cube number will form the shape of a cube.

## Cubing negative numbers

We can also cube negative numbers.
E.g.

\begin{array}{ll} (-5) \times(-5) \times(-5)=-125 & \quad \therefore(-5)^{3}=-125 \\\\ (-7) \times(-7) \times(-7)=-343 & \quad \therefore(-7)^{3}=-343 \end{array}

You will notice that when we cube a negative number we get a negative number.
This is because a negative number multiplied by a negative number multiplied by a negative number gives us a negative result.

When we cube negative 5 we get the negative answer of the cube of positive 5.
This is true for all numbers (and variables) and means:

$(5)^{3} \neq(-5)^{3}$

## What is a cube root?

The cube root of a number is a value that can be multiplied by itself three times to give the original number.
A cube root is the inverse operation of cubing a number.
The cube root function looks like this 3

When we cube a positive number we get a positive result and when we cube a negative number we get a negative result.
So the cube root of a positive number is also a positive number, and the cube root of a negative number is also a negative number.

E.g.

\begin{array}{l} \text { As } 3^{3}=27 \text { the cube root of } 27 \text { is } 3 & \quad \therefore \sqrt[3]{27}=3 \\\\ \text { As } 8^{3}=512 \text { the cube root of } 512 \text { is } 8 & \quad \therefore \sqrt[3]{512}=8 \\\\ \text { As} (-3)^{3}= -27 \text { the cube root of} -27 \text { is} -3 & \quad \therefore \sqrt[3]{-27}=-3 \end{array}

### Key words

Term

A single number (constant) or variable

E.g.
in the expression 4x − 7 both 4x and − 7 are terms

Coefficient

The number which the variable is being multiplied by

E.g.
in 2x3 the coefficient is 2

Integer

A whole number

E.g.
1, 7 or 1003

Index (also called exponent or powers)

The index number is the amount of times you multiply a number/variable by itself.

E.g.
the index number in 53 is 3

Note: the plural of index is indices

Note: you will see index number as a superscript

Base number

The number/unknown that is being multiplied by itself an amount of times

E.g.
the base number in 53 is 5 and in 2x3 the base number is x

Real numbers

Any positive or negative number is called a real number. Numbers that are not ‘real’ are called imaginary numbers. Integers, decimals, fractions are all examples of real numbers

E.g.
1, 2, 5 and 100 are all examples of a positive real numbers

-1, -2, -50 and -65.67 are all examples of a negative real number

Imaginary numbers

Numbers that are not real are called imaginary numbers, for example you will notice we cannot find the square root of a negative number (try it on a calculator), this because it is an imaginary number.. Numbers that contain an imaginary part and real part are called complex numbers.

## Cube numbers and cube roots examples

### Example 1

What is 5 cubed?

5 cubed means 5 × 5 × 5

So 5 cubed is 125

### Example 2

What is 9 cubed?

93 cubed means 9 × 9 × 9

So 9 cubed is 729

### Example 3

What is the cube root of 64?

The cube root of 64 means what value can be multiplied by itself three times to give 64

So

$\sqrt[3]{64} =4$

## How to use cube numbers and cube roots

1. Identify whether you need to cube or cube root the number/variable
2. Perform the operation
3. Clearly state the answer within the context of the question e.g. including units

Now you will focus on solving problems using your knowledge of cube numbers and cube roots.

## Cube numbers and cube roots problem examples

### Example 4: knowledge of cube numbers

Danny says 2 cubed is 6. Why is Danny wrong? What mistake did he make?

1. Identify whether you need to cube or cube root the number/variable

The question focus is on cubing 2 or “2 cubed”

2 Perform the operation

\begin{array}{l} 2 \times 2 \times 2 =2^{3} =8 \end{array}

3 Clearly state the answer within the context of the question

Danny is wrong because 2 cubed is 8 not 6.

The mistake Danny made was he did ‘2 × 3’ not ‘23.

### Example 5: problem solving with cube roots

Ava says the cube root of an integer is always smaller than the original number. Prove Ava is incorrect.

You are looking for a relationship between the cube root of an integer and the original number. It will help to write down your cube numbers and their cube roots.

\begin{array}{l} \sqrt[3]{1000}=10 \\\\ \sqrt[3]{729}=9 \\\\ \sqrt[3]{512}=8 \\\\ \sqrt[3]{343}=7 \\\\ \sqrt[3]{216}=6 \\\\ \sqrt[3]{125}=5 \\\\ \sqrt[3]{64}=4 \\\\ \sqrt[3]{27}=3 \\\\ \sqrt[3]{8}=2 \\\\ \sqrt[3]{1}=1 \end{array}

Ava is incorrect because

$\sqrt{1}=1$

The answer here is not smaller than the original number.

### Example 6: solving problems involving cube numbers

The sum of two cube numbers is 72. Find the two cube numbers.

Remember sum means add. Therefore you are looking for two cube numbers that add together to make 72.

It will help here to list the cube numbers up to 72

The Cube Numbers: 1     8     27     64

You now need to pick two of these numbers that when added together make 72

The two cube numbers are 64 and 8.

### Example 7: cube numbers within a 3D polygon

A cube has a volume of 125mm3. What is the length of one side?

The volume of a cube is found by multiplying the length, width and height together. For a cube the length, width and height are the same length.

Therefore you are looking for a number that when multiplied by itself three times (or cubed) is equal to 125.

Therefore you need to find the cube root of 125

$\sqrt[3]{125}=5$

Therefore the length of the cube is 5mm.

### Example 8: cube numbers within a 3D polygon

Length of one side of a cube is 4mm. What is the volume of the cube?

The volume of a cube is found by multiplying the length, width and height together. For a cube the length, width and height are the same length.

Therefore you are are going to multiply the side length by itself three or ‘cube’ the length.

Therefore you need to cube 4

${4^3}=64$

Therefore the volume of the cube is 64mm3.

### Example 9

Lexi says “when you add three consecutive cube numbers, the answer is always odd.”

To prove Lexi wrong we only need to find one example where she is incorrect, this is sometimes known as proof by contradiction.

Therefore we are looking for 3 consecutive cube numbers that when added together are even. It will be helpful here to list the ‘main’ cube numbers.

The cube numbers:

$1, \quad 8, \quad 27, \quad 64, \quad 125, \quad 216, \quad 343, \quad 512, \quad 729, \quad 1000$

You are now looking for one example where three of these numbers when added together make an even number.

For example: 1 + 8 + 27 = 36

Lexi is wrong because 1 + 8 + 27 = 36

### Common misconceptions

• Cube numbers

Incorrect understanding of cubing a number

E.g.
33 = 27 not 9

• Cube roots

Not recognising that a negative number cube rooted is negative

E.g.
\sqrt[3]-8 = -2

### Practice cube numbers and cube roots questions

1. What is 10 cubed?

30

100

1000

10000
10^{3}=10 \times 10 \times 10 = 1000

2. What is the value of the cube root of 216?

72

6

108

21
6 \times 6 \times 6 = 216 \quad \text{ therefore } \quad \sqrt[3]{216} = 6

3. What is (-7)^3?

343

-343

21

-21
(-7)^{3} = (-7) \times (-7) \times (-7) = -343

4. What is the value of \sqrt[3]{-1}?

0

1

3

-1
(-1) \times (-1) \times (-1) = -1 \quad \text{ therefore } \quad \sqrt[3]{-1}=-1

5. The sum of two cube numbers is 65 . Find the two cube numbers.

8 and 27

8 and 57

1 and 64

27 and 38

The cube numbers up to 65 are 1, 8, 27, 64.

The two that add up to 65 are 1 and 64.

6. A cube has a volume of 512 cm^3. What is the length of one side?

170.7cm

8cm

256cm

5.12cm

We find the volume of a cube by multiplying the length, the width and the height together.

These are all equal for a cube so we need to find a number that, when multiplied by itself three times, gives us 512.

Therefore we need to find the cube root of 512.

\sqrt[3]{512}=8

### Cube numbers and cube roots GCSE questions

1.  Work out the value of:

(a)    3^{3}

(b)    5 \times 2^{3}

(c)    6^{3} − 3^{3}

(3 Marks)

(a)    3 \times 3 \times 3

27

(1)

(b)    5 \times 8

40

(1)

(c)    216 – 27

189

(1)

2. Here is a list of numbers:

(a)   List the cube numbers

(b)   Which number is a square and cube number?

(2 Marks)

(a)    1000, 8 , 64 and 1

(1)

(b)    64

(1)

3. Find the value of:

(a)    \sqrt[3]{729}

(b)    \sqrt[3]{-1}

(c)    \sqrt[3]{-8}

(3 Marks)

(a)

(1)

(b)    -1

(1)

(c)    -2

(1)

4. Simplify the following expression

5^{3} \times \sqrt[3]{x^{3}}

(2 Marks)

125 x

Correct coefficient

(1)

Correct (or x^{1} )

(1)

## Learning checklist

You have now learned how to:

• Calculate cube numbers up to 10 × 10 × 10
• Use positive integer powers and their associated real roots
• Recognise and use the cube numbers
• Apply properties of cubes to a context

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