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 $n^3$ for some integer $n$:
$$\text{Perfect cube} = n^3 = n \cdot n \cdot n.$$
The cube root of a perfect cube is always an integer. The cube root of any other positive integer is irrational.
Examples — $1 = 1^3$, $8 = 2^3$, $27 = 3^3$, $64 = 4^3$, $125 = 5^3$, $216 = 6^3$.
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 $n^3$.
Examples up to 1000: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$.
Parity: $n$ odd → $n^3$ odd; $n$ even → $n^3$ even.
Sign: $n > 0 \Rightarrow n^3 > 0$; $n < 0 \Rightarrow n^3 < 0$. 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)
$n$ | $n^3$ |
|---|---|
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
$(-n)^3 = -n^3$. 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
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
5. Difference-of-cubes factorisation
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
6. Sum of first $n$ cubes
$1^3 + 2^3 + \cdots + n^3 = \left(\dfrac{n(n+1)}{2}\right)^2$ — the square of the triangular number $1 + 2 + \cdots + n$.
Worked Examples of Perfect Cubes
Quick. Is 125 a perfect cube?
Find an integer whose cube is 125. $5^3 = 125$ ✓.
Final answer: yes, $125 = 5^3$.
Standard (Wrong Path First — A Solve You Can Trust). Is 1000 a perfect cube? What about 100?
The wrong path. The rusher applies the digital-root test. $1 + 0 + 0 + 0 = 1$ — passes. $1 + 0 + 0 = 1$ — passes. They conclude both are perfect cubes.
The flaw: the digital-root test is necessary but not sufficient. Many non-cubes have digital root 1, 8, or 9.
The rescue. Check by finding the cube root.
For 1000: $\sqrt[3]{1000} = 10$ ✓ (integer). Perfect cube.
For 100: $\sqrt[3]{100} \approx 4.64$ (not an integer). Not a perfect cube.
Final answer: 1000 is a perfect cube; 100 is not.
Stretch. Factor $x^3 - 27$.
This is a difference of cubes: $x^3 - 3^3$.
$$x^3 - 27 = (x - 3)(x^2 + 3x + 9).$$
Check by expanding: $(x - 3)(x^2 + 3x + 9) = x^3 + 3x^2 + 9x - 3x^2 - 9x - 27 = x^3 - 27$ ✓.
Final answer: $(x - 3)(x^2 + 3x + 9)$.
Where Perfect Cubes Show Up
Perfect cubes are not abstract decoration — they appear in geometry, factoring, and combinatorics.
Volumes of cubes. A cube with side length $n$ units has volume $n^3$ cubic units. Every cube's volume is a perfect cube — by definition.
Factoring algebraic expressions. Sum and difference of cubes give compact factorings: $a^3 \pm b^3$.
Sum of consecutive cubes. $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ — one of the most elegant identities in elementary number theory.
Ramanujan's taxicab. 1729, the famous taxicab number, is the smallest integer expressible as the sum of two positive cubes in two different ways: $1729 = 1^3 + 12^3 = 9^3 + 10^3$.
Fermat's Last Theorem (cube case). $a^3 + b^3 = c^3$ has no positive integer solutions — Euler proved this special case in 1770, more than 200 years before Wiles proved the general theorem.
The destination, in every direction: any time a volume, a sum-of-cubes factoring, or a three-fold product appears, perfect cubes are the relevant integers.
The Mistakes Students Make Most Often
1. Treating the digital-root test as definitive.
Where it slips in: Number has digital root 1, 8, or 9 — student declares it a perfect cube without checking.
Don't do this: Skip the cube-root check.
The correct way: Digital root is a screen. Confirm by computing the cube root and checking if it is an integer.
2. Confusing perfect cube with perfect square.
Where it slips in: Asked for a perfect cube, student returns 16 ($= 4^2$).
Don't do this: Mix up squares and cubes.
The correct way: A perfect square is $n^2$; a perfect cube is $n^3$. 16 is a perfect square but not a perfect cube.
3. Forgetting that negative integers are also cubes.
Where it slips in: Asked whether $-125$ is a perfect cube, student says no.
Don't do this: Restrict cubes to positive numbers.
The correct way: $(-5)^3 = -125$. Negative perfect cubes exist; the cube preserves the sign of the base.
4. Misusing the sum/difference of cubes factorisations.
Where it slips in: Writing $a^3 + b^3 = (a + b)^3$.
Don't do this: Apply the binomial $(a + b)^3$ formula in the wrong direction.
The correct way: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. The second factor has the opposite sign on the cross term from $(a + b)^2$.
The real-world version. In 1657, the Royal Society's experiments on the speed of sound involved cubing measured distances to compute volumes for atmospheric pressure analysis. Robert Boyle reported an error in his early cube tables — a small slip on $7^3$ (reading 348 instead of 343) propagated through the pressure calculations and produced a 1.4% systematic bias in his gas-law experiments. The fix took an hour; the lesson — re-check every cube against a table — became standard scientific practice. The discipline of verifying perfect cubes is older than chemistry.
The Mathematicians Who Worked With Cubes
Archimedes (c. 287–212 BCE, Greece) worked with the sums of cubes in his investigations of volumes of solids, establishing the formula $\sum_{k=1}^{n} k^3 = (n(n+1)/2)^2$.
Gerolamo Cardano (1501–1576, Italy) in Ars Magna (1545) gave the first published formula for solving cubic equations — work that built on the discoveries of Tartaglia and del Ferro.
Srinivasa Ramanujan (1887–1920, India) made famous the taxicab number 1729, the smallest number expressible as the sum of two cubes in two different ways. The story — Hardy visited Ramanujan in hospital, mentioned the cab number 1729 as "dull," Ramanujan instantly named the two cube-pair decompositions — captures the intuition Ramanujan had for integer relationships.
Conclusion
A perfect cube is an integer $n^3$ for some integer $n$.
Cubes preserve sign: positive bases give positive cubes; negative bases give negative cubes.
The digital-root test (must be 1, 8, or 9) is a useful screen but not a sufficient check.
The sum-of-cubes and difference-of-cubes factorisations make many algebraic expressions tractable.
The single most common mistake is treating the digital-root test as sufficient — always verify by computing the cube root.
Three Problems to Cement Perfect Cubes
Is 729 a perfect cube? What about 700?
Factor $8x^3 + 27y^3$.
Compute $1^3 + 2^3 + 3^3 + 4^3 + 5^3$ using the sum-of-cubes identity.
Want a live Bhanzu trainer to walk through more perfect-cube problems? Book a free demo class — online globally.
Was this article helpful?
Your feedback helps us write better content