Loading Calculator...
Please wait a moment
Please wait a moment
Check if any number is a perfect cube, find its cube root if it is, list perfect cubes up to a given number, and handle negative perfect cubes with complete analysis.
Enter a number to check if it's a perfect cube
Type any number in the input field - positive or negative. Perfect cubes exist for both positive and negative numbers, unlike perfect squares which are only non-negative.
The calculator instantly determines if your number is a perfect cube. If yes, it shows the cube root. If no, it shows the nearest perfect cubes above and below.
Review the list of perfect cubes up to (or down to for negatives) your number to see the pattern and understand where your number falls in the sequence.
n³ = n × n × n
A perfect cube is a number that equals some integer multiplied by itself three times.
³√x ∈ ℤ ⟹ x is perfect cube
If the cube root is an integer, the number is a perfect cube.
1³ + 2³ + ... + n³ = (n(n+1)/2)²
Sum of first n cubes equals the square of the sum of first n numbers.
(-n)³ = -n³
Cubing a negative number gives a negative result. Example: (-3)³ = -27
a³ × b³ = (ab)³
Product of perfect cubes is a perfect cube. Example: 8 × 27 = 216
Last digit of n³ = last digit of n
Units digit of a perfect cube matches the units digit cubed. Example: 7³ = 343
| Number (n) | Perfect Cube (n³) | Calculation |
|---|---|---|
| -5 | -125 | -5 × -5 × -5 |
| -4 | -64 | -4 × -4 × -4 |
| -3 | -27 | -3 × -3 × -3 |
| -2 | -8 | -2 × -2 × -2 |
| -1 | -1 | -1 × -1 × -1 |
| 0 | 0 | 0 × 0 × 0 |
| 1 | 1 | 1 × 1 × 1 |
| 2 | 8 | 2 × 2 × 2 |
| 3 | 27 | 3 × 3 × 3 |
| 4 | 64 | 4 × 4 × 4 |
| 5 | 125 | 5 × 5 × 5 |
| 6 | 216 | 6 × 6 × 6 |
| 7 | 343 | 7 × 7 × 7 |
| 8 | 512 | 8 × 8 × 8 |
| 9 | 729 | 9 × 9 × 9 |
| 10 | 1000 | 10 × 10 × 10 |
A perfect cube is a number that can be expressed as an integer multiplied by itself three times. For example, 27 is a perfect cube because 27 = 3 × 3 × 3 = 3³. Perfect cubes are also called cube numbers. The sequence includes: ..., -8, -1, 0, 1, 8, 27, 64, 125, 216...
Take the cube root of the number. If the result is a whole number (integer), then it's a perfect cube. For example, ³√64 = 4 (integer), so 64 is a perfect cube. But ³√50 ≈ 3.68 (not integer), so 50 is not a perfect cube. This method works for both positive and negative numbers.
Yes! Unlike perfect squares, negative numbers can be perfect cubes. When you cube a negative number, you get a negative result. For example, (-3)³ = -27, so -27 is a perfect cube. Other negative perfect cubes include -1, -8, -64, -125, -216, etc.
Perfect squares result from squaring integers (n²), while perfect cubes result from cubing integers (n³). Key differences: (1) Perfect squares are always non-negative, but perfect cubes can be negative. (2) Perfect squares grow slower than perfect cubes. (3) Some numbers are both, like 64 = 8² = 4³.
Perfect cubes have interesting properties: (1) The last digit of n³ equals the last digit of n cubed. (2) The sum of first n cubes equals the square of their sum: 1³ + 2³ + ... + n³ = (1 + 2 + ... + n)². (3) Every integer is the sum of at most 9 perfect cubes (Waring's problem). (4) Product of perfect cubes is a perfect cube.
Perfect cubes appear in many contexts: (1) Geometry - volumes of cubes with integer side lengths. (2) Physics - cubic relationships in gas laws and planetary motion. (3) Number theory - Fermat's Last Theorem for n=3. (4) Computer science - hash functions and data structures. (5) Music - the circle of fifths involves cubic relationships.
Perfect cubes represent the volume of cubes with integer side lengths. A cube with side length n has volume n³, which is why these numbers are called "cubes." The ancient Greeks and Indians studied perfect cubes extensively, and they appear in many important mathematical theorems.
One remarkable property is that the sum of the first n cubes equals the square of the sum of the first n integers: 1³ + 2³ + 3³ + ... + n³ = (1 + 2 + 3 + ... + n)². This beautiful identity connects cubes to squares and triangular numbers, revealing deep patterns in number theory.