Perfect Cube Calculator
Enter any integer to find its cube (n cubed), check whether it is a perfect cube, and see the cube root. Switch to Cube Root mode to reverse-solve: type a result and find which integer was cubed to get it. The result updates instantly as you type, with a worked step panel showing the arithmetic.
What is a perfect cube?
A perfect cube is any integer that can be written as another integer raised to the power of 3. For example, 8 is a perfect cube because 2^3 = 2 x 2 x 2 = 8. The sequence of positive perfect cubes starts at 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 and continues indefinitely. Negative integers also produce perfect cubes: (-3)^3 = -27, so -27 is a perfect cube. Zero is included as well, since 0^3 = 0. The name comes from geometry: n^3 counts exactly how many unit cubes fit inside a solid cube with side length n, which makes the concept easy to visualise.
How to check whether a number is a perfect cube
To check whether an integer N is a perfect cube, compute its cube root and test whether the result is itself an integer. In practice: take the real cube root of N (preserving the sign for negatives), round the result to the nearest integer, and then cube that integer. If you recover N exactly, N is a perfect cube. For example, cube root of 343 is 7.000..., and 7^3 = 343, so 343 is a perfect cube. Cube root of 100 is approximately 4.6416, which rounds to 5 but 5^3 = 125 which does not equal 100, so 100 is not a perfect cube. This calculator performs that test instantly.
How to calculate n cubed
Cubing a number means multiplying it by itself twice: n^3 = n x n x n. For small integers this is straightforward: 4^3 = 4 x 4 x 4 = 64. For larger numbers, successive multiplication keeps precision: 15^3 = 15 x 15 x 15 = 225 x 15 = 3375. Negative bases follow the same rule, and since the exponent 3 is odd, the result is always negative: (-6)^3 = (-6) x (-6) x (-6) = 36 x (-6) = -216. Fractional and decimal bases can also be cubed, though the result is generally not a perfect cube.
Cube roots and how they relate to perfect cubes
The cube root of a number x is the value r such that r^3 = x. For any real number, there is exactly one real cube root (unlike square roots, which require non-negative input). The cube root of a perfect cube is always an integer: cube root of 8 = 2, cube root of 1000 = 10, cube root of -125 = -5. For all other integers, the cube root is irrational and must be approximated. The symbol used is the radical with a small 3, or equivalently x^(1/3) in exponential notation. This calculator computes cube roots to at least six decimal places for non-perfect cubes.
First 50 perfect cubes
| n | n^3 | n | n^3 |
|---|---|---|---|
| 1 | 1 | 26 | 17576 |
| 2 | 8 | 27 | 19683 |
| 3 | 27 | 28 | 21952 |
| 4 | 64 | 29 | 24389 |
| 5 | 125 | 30 | 27000 |
| 6 | 216 | 31 | 29791 |
| 7 | 343 | 32 | 32768 |
| 8 | 512 | 33 | 35937 |
| 9 | 729 | 34 | 39304 |
| 10 | 1000 | 35 | 42875 |
| 11 | 1331 | 36 | 46656 |
| 12 | 1728 | 37 | 50653 |
| 13 | 2197 | 38 | 54872 |
| 14 | 2744 | 39 | 59319 |
| 15 | 3375 | 40 | 64000 |
| 16 | 4096 | 41 | 68921 |
| 17 | 4913 | 42 | 74088 |
| 18 | 5832 | 43 | 79507 |
| 19 | 6859 | 44 | 85184 |
| 20 | 8000 | 45 | 91125 |
| 21 | 9261 | 46 | 97336 |
| 22 | 10648 | 47 | 103823 |
| 23 | 12167 | 48 | 110592 |
| 24 | 13824 | 49 | 117649 |
| 25 | 15625 | 50 | 125000 |
Every integer n from 1 to 50, its cube n^3, and whether the value is commonly recognised.
Frequently asked questions
Is 0 a perfect cube?
Yes. Zero is a perfect cube because 0^3 = 0 x 0 x 0 = 0. It is the only perfect cube that is neither positive nor negative.
Can negative numbers be perfect cubes?
Yes. Because cubing is an odd-exponent operation, negative integers produce negative results. For example, (-5)^3 = -125, so -125 is a perfect cube. This is different from square roots, where negative numbers have no real root.
Is 1 a perfect cube?
1 is a perfect cube because 1^3 = 1 x 1 x 1 = 1. It is the smallest positive perfect cube.
Is 64 a perfect cube?
64 is a perfect cube because 4^3 = 4 x 4 x 4 = 64. It is also a perfect square (8^2 = 64), making it one of a small group of numbers that are both.
What is the difference between a perfect cube and a perfect square?
A perfect square is an integer equal to n^2 for some integer n (e.g. 4, 9, 16). A perfect cube is an integer equal to n^3 for some integer n (e.g. 8, 27, 64). Some numbers qualify as both: 1, 64, 729, and 4096 are examples that are simultaneously perfect squares and perfect cubes, because they can be expressed as n^6.
How do I find cube roots without a calculator?
For small perfect cubes, memorising the list (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000) lets you recognise them on sight. For larger numbers, use prime factorisation: factor the number and check that every prime factor appears a multiple of 3 times. If 216 = 2^3 x 3^3, every exponent is a multiple of 3, so the cube root is 2 x 3 = 6. For non-perfect cubes, the digit-grouping method or Newton's method can approximate the root by hand.
How many perfect cubes are there between 1 and 1000?
There are exactly 10 positive perfect cubes between 1 and 1000 inclusive: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. Each corresponds to n^3 for n = 1 through 10.