# 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
1. Find the cube of 8.
2. Is 216 a cube number? If so, which base gives it?
3. Evaluate (−5)³.
4. Which cube number lies closest to 700?
5. Use the pattern to find the last digit of 12³ without full multiplication.
6. Find the value of 1³ + 2³ + 3³ + 4³ and show it is a perfect square.

### **Answers**
1. 8³ = 512.
2. Yes; 6³ = 216, so the base is 6.
3. (−5)³ = −125.
4. The closest is 729.
5. The base ends in 2, so the cube ends in 8.
6. 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.