Cube Number — Definition, Formula & Examples

A cube number (or perfect cube) is the result of multiplying a whole number by itself three times. For example, 8 is a cube number because

2 \times 2 \times 2 = 8

An integer $n$ is a perfect cube if there exists an integer $k$ such that $n = k^3$ . The set of positive perfect cubes is $\{1, 8, 27, 64, 125, 216, \ldots\}$ .

Key Formula

n = k^3 = k \times k \times k

Where:

$k$ = Any integer (the base being cubed)
$n$ = The resulting cube number

How It Works

To find a cube number, pick any integer and multiply it by itself twice more. For instance,

5^3 = 5 \times 5 \times 5 = 125

, so 125 is a perfect cube. Negative integers also produce cube numbers:

(-3)^3 = -27

, so

-27

is a perfect cube. The first ten positive cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. Geometrically, a cube number represents the volume of a cube whose side length is an integer.

Worked Example

Problem: Is 64 a cube number? If so, find which integer cubed gives 64.

Try small values: Test integers starting from 1:

1^3 = 1

2^3 = 8

3^3 = 27

4^3 = 64

4^3 = 4 \times 4 \times 4 = 64

Confirm: Since

4 \times 4 = 16

and

16 \times 4 = 64

, the calculation checks out.

Answer: Yes, 64 is a cube number because

4^3 = 64

Visualization

Why It Matters

Cube numbers appear whenever you calculate the volume of a cube-shaped object, such as finding how many unit cubes fit inside a box. Recognizing perfect cubes also helps you simplify cube roots in algebra and pre-algebra courses.

Common Mistakes

Mistake: Confusing cubing with multiplying by 3. Students write

4^3 = 12

instead of

4^3 = 64

Correction: Cubing means multiplying the number by itself three times:

4^3 = 4 \times 4 \times 4

, not

4 \times 3