Perfect Cube — Definition, List, Examples
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
- Treating the digital-root test as definitive.
- Don't skip the cube-root check.
- Confusing perfect cube with perfect square.
- A perfect square is n²; a perfect cube is n³.
- Forgetting negative integers can also be cubes.
- Negative perfect cubes exist.
- 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
- Is 729 a perfect cube? What about 700?
- Factor 8x³ + 27y³.
- Compute 1³ + 2³ + 3³ + 4³ + 5³ using the sum-of-cubes identity.