# Perfect Cube — Definition, List, Examples

## TL;DR
A perfect cube is an integer of the form n³ where n is an integer — like 1, 8, 27, 64, 125. This article covers the definition, the list of perfect cubes from 1 to 1000, three worked examples of identifying perfect cubes, the digital-root test, and how perfect cubes connect to volumes and to factoring sums and differences of cubes.

## Numbers Built From a Triple Multiplication
Some numbers are special because of _how_ they are built. A perfect cube is built by multiplying an integer by itself, then by itself again. The result is always exactly three layers of the same factor — never two, never four.

Perfect cubes are the volume side of the family — perfect squares describe areas of squares, perfect cubes describe volumes of cubes. The geometric link is what makes them more than a school exercise.

## What a Perfect Cube Is
A **perfect cube** is an integer that can be written as n³ for some integer n:

Perfect cube = n³ = n ⋅ n ⋅ n.

The cube root of a perfect cube is always an integer. The cube root of any other positive integer is irrational.

Examples: 1 = 1³, 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³, 216 = 6³.

Perfect cubes can also be negative: (−3)³ = −27, (−5)³ = −125. Unlike squares (which are always non-negative), cubes preserve the sign of the base.

> **Quick Facts:**
> - **Definition:** integer of the form n³.
> - **Examples up to 1000:** 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
> - **Parity:** n odd → n³ odd; n even → n³ even.
> - **Sign:** n > 0 ⇒ n³ > 0; n < 0 ⇒ n³ < 0. Cubes preserve sign.
> - **Digital root:** always 1, 8, or 9 (a useful but not exhaustive test).
> - **Grade introduced:** CBSE Class 7–8 (exponents, cubes, cube roots); CCSS-M 8.EE.A.2 (evaluate square roots and cube roots of small perfect cubes); NCERT Class 8 Chapter 6 — Squares and Square Roots / Chapter 7 Cubes and Cube Roots.

## The List of Perfect Cubes (1 to 20)

| n  | n³   |
|----|------|
| 1  | 1    |
| 2  | 8    |
| 3  | 27   |
| 4  | 64   |
| 5  | 125  |
| 6  | 216  |
| 7  | 343  |
| 8  | 512  |
| 9  | 729  |
| 10 | 1000 |
| 11 | 1331 |
| 12 | 1728 |
| 13 | 2197 |
| 14 | 2744 |
| 15 | 3375 |
| 16 | 4096 |
| 17 | 4913 |
| 18 | 5832 |
| 19 | 6859 |
| 20 | 8000 |

## Properties of Perfect Cubes

### **1. Sign preservation**
(−n)³ = −n³. The cube of a negative integer is negative.

### **2. Parity**
Even × even × even = even; odd × odd × odd = odd. A perfect cube has the same parity as its cube root.

### **3. Digital-root test**
Sum the digits repeatedly until a single digit remains. The digital root of any perfect cube is 1, 8, or 9. _Necessary but not sufficient_ — not every number with digital root 1, 8, or 9 is a perfect cube.

### **4. Sum-of-cubes factorisation**
a³ + b³ = (a + b)(a² - ab + b²).

### **5. Difference-of-cubes factorisation**
a³ - b³ = (a - b)(a² + ab + b²).

### **6. Sum of first n cubes**
1³ + 2³ + ⋯ + n³ = (n(n + 1) / 2)².

## Worked Examples of Perfect Cubes
**Quick.** Is 125 a perfect cube?
Find an integer whose cube is 125. 5³ = 125 ✓.
**Final answer:** yes, 125 = 5³.

**Stretch.** Factor x³ − 27.
This is a difference of cubes: x³ − 3³.

x³ − 27 = (x - 3)(x² + 3x + 9). Check by expanding: (x − 3)(x² + 3x + 9) = x³ − 27 ✓.

## Where Perfect Cubes Show Up
Perfect cubes appear in geometry, factoring, and combinatorics.
- **Volumes of cubes.** A cube with side length n units has volume n³ cubic units.
- **Factoring algebraic expressions.** Sum and difference of cubes give compact factorings.
- **Sum of consecutive cubes.** One of the most elegant identities in elementary number theory.
- **Fermat's Last Theorem (cube case).** a³ + b³ = c³ has no positive integer solutions.

## The Mistakes Students Make Most Often
1. Treating the digital-root test as definitive.
   - Don't skip the cube-root check.
2. Confusing perfect cube with perfect square.
   - A perfect square is n²; a perfect cube is n³.
3. Forgetting negative integers can also be cubes.
   - Negative perfect cubes exist.
4. Misusing the sum/difference of cubes factorisations.
   - The cross term sign is opposite.

## The Mathematicians Who Worked With Cubes
- **Archimedes (c. 287–212 BCE)** worked with the sums of cubes.
- **Gerolamo Cardano (1501–1576)** published the first formula for solving cubic equations.
- **Srinivasa Ramanujan (1887–1920)** made famous the taxicab number 1729.

## Conclusion
- A **perfect cube** is an integer n³ for some integer n.
- Cubes preserve sign: positive bases give positive cubes; negative bases give negative cubes.
- Always verify by computing the cube root.

## Three Problems to Cement Perfect Cubes
1. Is 729 a perfect cube? What about 700?
2. Factor 8x³ + 27y³.
3. Compute 1³ + 2³ + 3³ + 4³ + 5³ using the sum-of-cubes identity.