Cube Numbers — Definition, List, and Examples
Cube Numbers — Definition, List, and Examples
What Is a Cube Number?
A cube number is the result of multiplying a whole number by itself twice more — three copies in total — written as n×n×n, or n³. So 2³ = 2 × 2 × 2 = 8, and 8 is a cube number.
The first cube numbers are:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Cube Numbers From 1 to 10
Every cube number comes from a base integer. Here is the full list from 1³ to 10³, the range most students are asked to know by heart.
| n | n³ | Read as |
|---|---|---|
| 1 | 1 | one cubed |
| 2 | 8 | two cubed |
| 3 | 27 | three cubed |
| 4 | 64 | four cubed |
| 5 | 125 | five cubed |
| 6 | 216 | six cubed |
| 7 | 343 | seven cubed |
| 8 | 512 | eight cubed |
| 9 | 729 | nine cubed |
| 10 | 1000 | ten cubed |
Notice how fast they grow. The gap from 1 to 8 is 7; the gap from 9³ to 10³ is 271. Cubes pull away from each other far quicker than square numbers do, because each step multiplies by an extra factor of n.
Is a cube number the same as a perfect cube?
Yes. "Cube number" and "perfect cube" name the same thing: an integer that is some whole number multiplied by itself three times.
How to Find a Cube Number
Finding a cube number takes one rule: multiply the base by itself, then by itself once more. There are two reliable ways to do it.
- Direct multiplication. Write the base three times and multiply left to right, one step at a time.
- Square-then-multiply. Square the base, then multiply by the base once more.
Where Cube Numbers Come From
Cube numbers are one branch of the exponents family. Raising a number to the power 3 is called cubing it, and it sits one step beyond squaring.
- Squaring (n²) measures the area of a square with side n.
- Cubing (n³) measures the volume of a cube with edge n.
There is a second pattern hiding in the list. Add up consecutive odd numbers and cube numbers appear:
1 = 1 = 1 = 1 + 3 + 5 = 8 + 5 = 8 + 9 + 11 = 27 + 9 + 11 = 27 + 13 + 15 + 17 + 19 = 64.
Example 1: Find the cube of 6
6³ = 6 × 6 × 6 = 216. Final answer: 6³ = 216.
Example 2: Find the cube of 5
5³ = 5 × 5 × 5 = 125. Final answer: 5³ = 125.
Example 3: Is 100 a cube number?
100 sits between 64 and 125, and there is no integer between 4 and 5. Final answer: 100 is not a cube number.
Example 4: Find the cube of the negative number −3
(−3)³ = (−3) × (−3) × (−3) = −27. Final answer: (−3)³ = −27.
Example 5: Which cube number is closest to 500?
List the cubes around 500: 7³ = 343 and 8³ = 512, compare the distances. Final answer: 512 is the closest cube number to 500.
Example 6: A storage crate is a perfect cube. Its volume is 729 cubic centimeters. How long is each edge?
The volume of a cube is edge cubed, so you need the base number whose cube is 729. Final answer: each edge is 9 cm long.
Properties of Cube Numbers
Cube numbers follow a handful of patterns that make them easy to recognize and check.
- Odd stays odd, even stays even. The cube of an even number is even; the cube of an odd number is odd.
- Negative bases give negative cubes. Three negative factors multiply to a negative.
- The unit digit is predictable. A cube's last digit is fixed by the base's last digit.
| Last digit of n | Last digit of n³ |
|---|---|
| 0, 1, 4, 5, 6, 9 | same digit |
| 2 | 8 |
| 8 | 2 |
| 3 | 7 |
| 7 | 3 |
- Each cube is a run of consecutive odd numbers.
- The sum of the first n cubes is a perfect square.
Common Mistakes With Cube Numbers
Mistake 1: Treating "cubed" as "multiply by 3"
Don't do this: writing 5³ = 15. The correct way: cubing is three copies multiplied.
Mistake 2: Multiplying only twice
Don't do this: computing 4³ as 4 × 4 = 16 and stopping. The correct way: carry the third factor.
Mistake 3: Confusing a cube with a cube root
Don't do this: cubing 64 to get a huge number when the question wanted the base. The correct way: read the direction.
Practice Questions
- Find the cube of 8.
- Is 216 a cube number? If so, which base gives it?
- Evaluate (−5)³.
- Which cube number lies closest to 700?
- Use the pattern to find the last digit of 12³ without full multiplication.
- Find the value of 1³ + 2³ + 3³ + 4³ and show it is a perfect square.
Answers
- 8³ = 512.
- Yes; 6³ = 216, so the base is 6.
- (−5)³ = −125.
- The closest is 729.
- The base ends in 2, so the cube ends in 8.
- 1 + 8 + 27 + 64 = 100 = 10².
Conclusion
- A cube number is an integer multiplied by itself three times: n³ = n × n × n.
- The first ten cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- Each cube n³ is the volume of a cube with edge n, and equals a run of n consecutive odd numbers.