Perfect Cube Calculator (is it a perfect cube? cube root & nearest cubes)

Understanding the perfect cube calculator — concept and principle

This calculator tests whether a whole number is a perfect cube, gives its cube root, simplifies the cube root when it is not whole, and shows the nearest perfect cubes below and above. Use it for algebra and number-theory exercises, for simplifying radicals, or for checking volumes of cubes with whole-number sides.

3,375 small cubes fill a 15 × 15 × 15 block exactly, so 3,375 is a perfect cube.

How it is calculated

n is a perfect cube if n = k³ for some whole number k.
The calculator finds the largest whole number k with k³ ≤ n, using exact integer arithmetic, and checks whether k³ = n.
If n is not a cube, it factors n into primes, takes out the largest cube factor and writes ∛n = k·∛m.
It also lists k³ (the largest cube ≤ n), (k + 1)³ (the next cube) and the distance to the nearer one.

Negative numbers are allowed because (−k)³ = −k³.

The principle

Cubing a whole number triples every exponent in its prime factorisation. So a number is a perfect cube exactly when every prime exponent is a multiple of 3. For example 3,375 = 3³ × 5³ = (3 × 5)³ = 15³. When the exponents are not all multiples of 3, the cube-number part can still be pulled out of the root: ∛54 = ∛(27 × 2) = 3∛2. Perfect cubes grow quickly and get further apart: the gap between k³ and (k + 1)³ is 3k² + 3k + 1.

Worked example

Default input: n = 3,375.

∛3,375 = 15, and 15³ = 15 × 15 × 15 = 3,375 exactly.
Result: Yes, a perfect cube: 3,375 = 15³.
Prime factorisation: 3,375 = 3³ × 5³, all exponents are 3.
The next cube is 16³ = 4,096; the gap is 3(15²) + 3(15) + 1 = 721.
Distance to the nearest perfect cube = 0, and the simplified radical is just 15.

How to read the result

"Yes" means the cube root is a whole number. "No" means the cube root is irrational, and the simplified radical is the exact form to use in algebra. The nearest cubes help with estimates: a number between 3,375 and 4,096 has a cube root between 15 and 16.

Common mistakes

Testing with a rounded decimal cube root; floating-point results like 14.9999999 can mislead. The calculator uses whole-number checks.
Thinking negative numbers cannot be cubes. −27 = (−3)³.
Confusing perfect cubes with perfect squares; 64 is both (4³ and 8²), but 27 is only a cube.
Simplifying with a cube factor that is not the largest one.

Keep in mind: Inputs must be whole numbers between −10¹⁵ and 10¹⁵. Fractions and decimals are not tested as cubes of rational numbers.

Frequently asked questions

What are the first ten perfect cubes?

1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000.

Is 0 a perfect cube?

Yes, 0 = 0³.

How can I tell quickly if a number might be a cube?

Cubes of 0–9 end in 0, 1, 8, 7, 4, 5, 6, 3, 2 and 9, so the last digit alone never rules a number out, but the digital root of a positive cube is always 1, 8 or 9.

Sources: OpenStax, Intermediate Algebra 2e (2020), §8.3 Simplify Rational Exponents and §8.5

Checked 9 Nov 2026. OpenStax, Intermediate Algebra 2e (2020), §8.3 Simplify Rational Exponents (a^(1/n) = ⁿ√a; the denominator of the exponent is the index of the root) and §8.5 (a radicand that is a perfect power of the index comes out of the root). Perfect cube k³ and simplification ∛(k³m) = k∛m — standard definitions.

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