Cube Numbers and Cube Roots

What does cubing a number mean?

You already know that multiplying a number by itself is called squaring. Cubing goes one step further: you multiply a number by itself three times.

We write a cube using a small raised 3, called an index or power. So "four cubed" is written 4³, and it means:

4³ = 4 × 4 × 4 = 64

The word cube is no accident. If you build a solid cube out of small bricks that is 4 bricks wide, 4 bricks deep and 4 bricks tall, you need exactly 4 × 4 × 4 = 64 bricks. That is why the answer is called a cube number.

The first cube numbers

Let's calculate the first few cube numbers. Take each whole number and multiply it by itself three times.

Number (n)	Working (n × n × n)	Cube number (n³)
1	1 × 1 × 1	1
2	2 × 2 × 2	8
3	3 × 3 × 3	27
4	4 × 4 × 4	64
5	5 × 5 × 5	125
6	6 × 6 × 6	216
10	10 × 10 × 10	1000

Notice how quickly cube numbers grow — much faster than square numbers. By the time you reach 10, the cube is already 1000.

Worked example: finding a cube

Example — What is 6³?

1. Write it out: 6³ = 6 × 6 × 6.
2. Multiply the first two: 6 × 6 = 36.
3. Multiply by the last 6: 36 × 6 = 216.

So 6³ = 216. Always do the multiplication in steps — never multiply by 3.

What is a cube root?

A cube root is the opposite, or inverse, of cubing. It asks: "Which number, when cubed, gives this answer?" We use the symbol ∛.

Because 3 × 3 × 3 = 27, the cube root of 27 is 3. We write this as ∛27 = 3.

Example — Find ∛64.

1. Ask: what number cubed makes 64?
2. Try 4: 4 × 4 × 4 = 64. It works.
3. So ∛64 = 4.

If a number has a whole-number cube root (like 1, 8, 27, 64, 125), we call it a perfect cube. Learning the first few cubes by heart makes finding cube roots fast.

Cubes and negative numbers

Unlike squares, cubes can be negative. Multiplying three negative numbers gives a negative result:

(-3)³ = (-3) × (-3) × (-3) = 9 × (-3) = -27

So the cube root of -27 is -3. (You can revisit the sign rules in operations with negative numbers.)

Where cube numbers are used

The most common place you meet cubes is volume. The volume of a cube is simply the side length cubed:

Volume = side × side × side = side³

A storage box with sides of 3 m holds 3³ = 27 cubic metres. Cubic units like cm³ and m³ are everywhere in measuring space, which is exactly why the operation is named after the cube shape.

A practice activity

Try this "cube hunt" with a partner:

1. Write the cube numbers 1, 8, 27, 64 and 125 on cards.
2. Shuffle a second set of cards showing 1, 2, 3, 4 and 5.
3. Take turns matching each number card to its cube. Say the rule aloud: "3 cubed is 27."
4. Challenge: build a cube tower from sugar cubes or building bricks that is 3 wide, 3 deep and 3 tall. Count the bricks — you should use exactly 27.

Where this leads

Cube numbers and cube roots prepare you for working with volume, powers and exponents, and prime factorisation. Memorise the first five cubes — 1, 8, 27, 64, 125 — and you will spot them instantly in harder problems later.