Cube Numbers - GCSE Maths Definition

Definition

A cube number is the result when you multiply a natural number (a positive integer (whole) number) by itself, and then multiply by itself again.

Explanation

Cube numbers are brilliant mathematical patterns you'll spot everywhere. They're called "cube" numbers because they are the volumes of cubes with the side length of where they come from. For example, a cube of side length 4, will have volume 4 x 4 x 4 = 64 cubic units.

Think of it this way. When you have a number like 4, you can cube it by doing 4 × 4 × 4. This gives you 64, which is a cube number.

We write cube numbers using powers. In 4³, 4 is the base number and 3 is the power. We would read or say this as "4 cubed" or "4 to the power of 3."

The first few cube numbers are

• 1³ = 1 × 1 × 1 = 1

• 2³ = 2 × 2 × 2 = 8

• 3³ = 3 × 3 × 3 = 27

• 4³ = 4 × 4 × 4 = 64

• 5³ = 5 × 5 × 5 = 125

It will help if you can remember the first few cube numbers, and other common ones, such as 10³ = 10 x 10 x 10 = 1000. These numbers pop up loads in maths. You'll see them in algebra, geometry, and even in real-world problems about volume.

Here's something cool. If you arrange small cubes into a bigger cube shape, the total number of small cubes is always a cube number. A 3×3×3 cube contains exactly 27 smaller cubes.

Cube numbers grow really quickly. While 1³ = 1, by the time you get to 10³, you're already at 1000.

Although we commonly use “the” cube numbers, as a mathematical operation you can cube any number. When you cube a negative number, you get a negative result. For example, (-2)³ = (-2) × (-2) × (-2) = -8.

The opposite of cubing is finding the cube root. If 2³ = 8, then the cube root of 8 is 2. We write this as ∛8 = 2.

Example

Question:
(a) Calculate the value of 6³.
(b) Write down the volume of a cube that has side length 6 cm.

Solution:
(a) Calculate 6³, 6³ means 6 × 6 × 6

Let's work this out step by step:

• First: 6 × 6 = 36

• Then: 36 × 6 = 216

Answer: 6³ = 216

(b) The volume of a cube is found by cubing its side length - this has been done in part (a)

Answer: Volume = 216 cm³

Common mistakes (and how to avoid them)

Mistake 1: Confusing cubes with squares

Many students mix up square numbers (like 4²) with cube numbers (like 4³). This leads to wrong calculations.

4² = 4 × 4 = 16 (square number)
4³ = 4 × 4 × 4 = 64 (cube number)

How to avoid this: Always check the power. If it's 2, you're squaring. If it's 3, you're cubing.

Mistake 2: Mixing up cube numbers and cube roots

Some students confuse finding a cube number with finding a cube root.

If you're asked "What is 5³?" the answer is 125. If you're asked "What is ∛125?" the answer is 5.

How to avoid this: Read the question carefully. Look for the cube symbol (³) or the cube root symbol (∛). They're asking for opposite operations.

Mistake 3: Getting the signs wrong with negative numbers

Students often struggle when cubing negative numbers. They forget that negative × negative × negative = negative.

For example: (-3)³ = (-3) × (-3) × (-3) = -27

How to avoid this: Remember that when you multiply an odd number of negative numbers together, the result is negative.

Mistake 4: Calculation errors in mental maths

Cube numbers get big quickly, and it's easy to make arithmetic mistakes when working them out.

How to avoid this: Break down the calculation. Instead of trying to do 7 × 7 × 7 in your head all at once, do 7 × 7 = 49 first, then multiply 49 × 7. Use a calculator when the numbers get tricky.

Mistake 5: Not recognising common cube numbers

Students sometimes don't spot cube numbers in exam questions, missing chances to simplify their work.

How to avoid this: Learn the first ten cube numbers by heart. Know that 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000 are all cube numbers. This helps you recognise them instantly.

Frequently asked questions

What's the difference between 8² and 8³?

8² = 8 × 8 = 64 (this is "8 squared")
8³ = 8 × 8 × 8 = 512 (this is "8 cubed")

The power tells you how many occurrences of the base number need multiplying.

Can you have cube numbers with decimals?

Yes, absolutely. For example, (0.5)³ = 0.5 × 0.5 × 0.5 = 0.125. Any number can be cubed, including decimals and fractions, but these are not included when we refer to “the” cube numbers.

What's the cube root of a negative number?

You can find cube roots of negative numbers. For example, ∛(-27) = -3 because (-3)³ = -27. This is different from square roots, where you can't have negative numbers under the root sign.

How do cube numbers relate to volume?

Cube numbers give you the volume of a cube with side length of that number. If a cube has sides of length 4 cm, its volume is 4³ = 64 cm³. That's why we measure volume in "cubic" units like cm³ or m³.

Do I need to memorise cube numbers for GCSE?

It's really helpful to know the first few cube numbers (1, 8, 27, 64, 125) off by heart. This speeds up your calculations and helps you spot patterns in exam questions. Remembering other common/easy cube numbers such as 10³ = 1000 is also helpful.

Related GCSE Mathematics glossaries

Cube number