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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.

A **cube number **is the product of **three identical factors**. To find a cube number, multiply an integer (whole number) by **itself **and then **itself again**, for example, 3 x 3 x 3 = 27. These are sometimes called *‘perfect cubes’*.

We can cube numbers with decimals but we do not refer to these as cube numbers.

A given number / variable that is* ‘ cubed’* is multiplied by itself three times.

E.g. ^{3}*“4** cubed* or

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

Here are the first

\begin{align*}
1 \times 1 \times 1 &=1 \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 1 \text { is a cube number } \\\\
2 \times 2 \times 2&=8 \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 8 \text { is a cube number } \\\\
3 \times 3 \times 3&=27 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 27 \text { is a cube number } \\\\
4 \times 4 \times 4&=64 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 64 \text { is a cube number } \\\\
5 \times 5 \times 5&=125 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 125 \text { is a cube number } \\\\
6 \times 6 \times 6&=216 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 216 \text { is a cube number } \\\\
7 \times 7 \times 7&=343 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 343 \text { is a cube number } \\\\
8 \times 8 \times 8&=512 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 512 \text { is a cube number } \\\\
9 \times 9 \times 9&=729 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 729 \text { is a cube number } \\\\
10 \times 10 \times 10&=1000 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 1000 \text { is a cube number }
\end{align*}

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 ^{3}

Any cube number will form the shape of a cube.

The cube number of an integer is also called a perfect cube.

We can cube numbers with decimal places but we do not refer to these as cube numbers or perfect cubes.

A given number or variable that is* ‘cubed’* is multiplied by itself and this applies to decimals as to integers.

It’s relatively straightforward to work out the cube root of a perfect cube but it is a much harder process to work out decimal cube roots. You quickly end up working with surds and irrational 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.**

Learn more by reviewing our lesson on negative numbers.

When we cube negative

This is true for all numbers (and variables) and means:

\[(5)^{3} \neq(-5)^{3}\]

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 symbol 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}

Get your free cube numbers worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free cube numbers worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Cube numbers and cube roots** 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:

**Term**

A single number (constant) or variable

E.g.

in the expression

**Coefficient**

The number which the variable is being multiplied by

E.g.

in ^{3}

**Integer**

A whole number

E.g.

**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 ^{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 ^{3 }^{3}

**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.

**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.

What is

So

What is

^{3}

So

What is the cube root of

The cube root of

So

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

- Identify whether you need to cube or cube root the number/variable
- Perform the operation
- 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.

Danny says

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

The question focus is on cubing

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

The mistake Danny made was he did ^{3}’

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

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

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.

Perform the operation

\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}

Clearly state the answer within the context of the question

Ava is incorrect because

\[\sqrt{1}=1\]

The answer here is not smaller than the original number.

The sum of two cube numbers is

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

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

It will help here to list the cube numbers up to

Perform the operation

The Cube Numbers:

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

Clearly state the answer within the context of the question

The two cube numbers are

A cube has a volume of ** ^{3}**. What is the length of one side?

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

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

Therefore you need to find the cube root of

Perform the operation

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

Clearly state the answer within the context of the question

Therefore the length of the cube is

Length of one side of a cube is

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

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

Perform the operation

\[{4^3}=64\]

Clearly state the answer within the context of the question

Therefore the volume of the cube is ** ^{3}**.

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

Is Lexi right? Explain your answer.

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

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

Perform the operation

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:

Clearly state the answer within the context of the question

Lexi is wrong because

**Cube numbers**

Incorrect understanding of cubing a number

E.g. ^{3} = 27

**Cube roots**

Not recognising that a negative number cube rooted is negative

E.g.

\sqrt[3]-8 = -2

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

1. Work out the value of:

(a) 3^{3}

(b) 5 \times 2^{3}

(c) 6^{3} − 3^{3}

**(3 Marks)**

Show answer

(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:

1000 \quad \quad 18 \quad \quad 8 \quad \quad 64 \quad \quad 7 \quad \quad 144 \quad \quad 1 \quad \quad 19

(a) List the cube numbers

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

**(2 Marks)**

Show answer

(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)**

Show answer

(a) 9

**(1)**

(b) -1

**(1)**

(c) -2

**(1)**

4. Simplify the following expression

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

**(2 Marks)**

Show answer

125 x

Correct coefficient

**(1)**

Correct x (or x^{1} )

**(1)**

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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