Why "cube"?
A cube number is a number multiplied by itself three times. Picture building a cube from small blocks: 4 blocks long, 4 wide and 4 tall needs 4 × 4 × 4 = 64 blocks. The volume of that cube is 64 cubic units.
Notation
4 × 4 × 4 is written 4³, read "four cubed". The exponent 3 tells you to use 4 as a factor three times.
The first perfect cubes
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1,000 |
A perfect cube has an integer cube root: ∛64 = 4, so 64 is a perfect cube.
Negative cubes
Unlike squares, cubes can be negative:
- (−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8
A negative number cubed stays negative. That’s why ∛(−27) = −3 makes sense.
Patterns
- The last digit of a cube identifies the last digit of its root uniquely: cubes ending in 8 come from numbers ending in 2 (2³ = 8, 12³ = 1,728).
- In a prime factorization of a perfect cube, every exponent is a multiple of 3: 216 = 2³ × 3³.
Cubes of tens and decimals
- 20³ = 8,000
- 0.1³ = 0.001
- 0.5³ = 0.125
Squares vs cubes
| Square | Cube | |
|---|---|---|
| Meaning | n × n | n × n × n |
| Shape | flat square (area) | solid cube (volume) |
| Units | cm² | cm³ |
| 3 → | 9 | 27 |
| −3 → | 9 | −27 |
Worked examples
Example 1: What is 4³?
4 × 4 × 4 = 16 × 4 = 64.
Example 2: A cube-shaped box has edges of 6 cm. What is its volume?
6³ = 216 cm³.
Example 3: Is 100 a perfect cube?
4³ = 64 and 5³ = 125. 100 is in between, so no.
Common mistakes
- Multiplying by 3 instead of cubing: 4³ is 64, not 12.
- Stopping after two factors: 4³ is not 16.
- Thinking a negative cubed is positive.
Try it yourself
- 3³
- 10³
- (−5)³
- A cube has volume 343 cm³. What is its edge length?
- Is 1,000,000 a perfect cube?
Answers: 1) 27 2) 1,000 3) −125 4) 7 cm 5) yes (100³)
Quick recap
- A cube number is n × n × n, written n³.
- Perfect cubes have integer cube roots.
- Cubes give volume, and negative numbers cubed stay negative.