Cubic Number — Definition, Formula & Examples

A cubic number is the result you get when you multiply an integer by itself three times. For example, 8 is a cubic 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 non-negative perfect cubes is $\{0, 1, 8, 27, 64, 125, 216, \ldots\}$ .

Key Formula

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

Where:

$k$ = Any integer (positive, negative, or zero)
$n$ = The resulting cubic number (perfect cube)

How It Works

To check whether a number is a perfect cube, try to find an integer whose third power equals that number. Negative integers also produce cubic numbers:

(-3)^3 = -27

, so

-27

is a perfect cube. The first several positive perfect cubes are

1^3 = 1

2^3 = 8

3^3 = 27

4^3 = 64

5^3 = 125

, and

6^3 = 216

. Memorizing these values helps you work faster with exponents and roots.

Worked Example

Problem: Is 125 a cubic number?

Step 1: Test integers by cubing them. Try

k = 5

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

Step 2: Since

5^3

equals 125 exactly, with no remainder, 125 is a perfect cube.

Answer: Yes, 125 is a cubic number because

125 = 5^3

Why It Matters

Recognizing perfect cubes lets you simplify cube roots quickly, which comes up in algebra when solving equations like

x^3 = 64

. Volume calculations in geometry also produce cubic numbers — the volume of a cube with side length 4 is

4^3 = 64

cubic units.

Common Mistakes

Mistake: Confusing cubic numbers with square numbers. For example, thinking

2^3 = 6

instead of

2^3 = 8

Correction: Squaring means multiplying a number by itself twice (

2^2 = 4

). Cubing means multiplying it by itself three times (

2^3 = 2 \times 2 \times 2 = 8

). Count the factors carefully.