The cube root undoes cubing: if a cube-shaped box holds 1,000 cubic inches, its edge is ∛1,000 = 10 inches. Cube roots are friendlier than square roots in one important way: every real number, positive or negative, has exactly one real cube root. This calculator finds it exactly when the number is a perfect cube, in simplest radical form when it is not, and always as a decimal.
How to use the cube root calculator
- Enter the radicand: a whole number, a decimal such as 0.001, a fraction such as 1/8, or a negative number.
- Read the cube root on the tape. Perfect cubes give an exact answer; other numbers show a simplified radical like 3∛2 and a ten-place decimal.
- Follow the steps to see the prime factorization and which factors come out of the radical.
Cube root formulas
| Rule | Formula | Example |
|---|---|---|
| Product | ∛(ab) = ∛a · ∛b | ∛54 = ∛27 · ∛2 = 3∛2 |
| Quotient | ∛(a/b) = ∛a / ∛b | ∛(1/8) = 1/2 |
| Negative radicand | ∛(−a) = −∛a | ∛−54 = −3∛2 |
| Rationalizing | 1/∛a = ∛(a²)/a | 1/∛2 = ∛4/2 |
Worked example: ∛54
Factor: 54 = 2 × 3³.
Group in threes: the three 3s form a complete group and leave the radical as a single 3; the lone 2 stays inside.
Result: ∛54 = 3∛2 ≈ 3 × 1.2599210499 = 3.7797631497.
Check: 3³ = 27 and 4³ = 64, and 54 is between them, so the root is between 3 and 4. ✓
With a negative sign, the same work gives ∛−54 = −3∛2 ≈ −3.7797631497. Decimals that are perfect cubes come out exactly: ∛0.001 = 0.1 because 0.1 × 0.1 × 0.1 = 0.001.
Common cube roots
| x | ∛x | x | ∛x |
|---|---|---|---|
| 1 | 1 | 216 | 6 |
| 8 | 2 | 343 | 7 |
| 27 | 3 | 512 | 8 |
| 64 | 4 | 729 | 9 |
| 125 | 5 | 1,000 | 10 |
| 2 | ≈ 1.2599 | 3 | ≈ 1.4422 |
| 10 | ≈ 2.1544 | 100 | ≈ 4.6416 |
Real-world uses
- Scaling volumes. If you want a cube-shaped planter that holds twice as much soil, each edge grows by ∛2 ≈ 1.26, only 26% longer, not twice as long.
- Finding a side from a volume. A cubic water tank of 8 m³ has edges of ∛8 = 2 m.
- Density and size. Because mass grows with volume, a sphere with 8 times the mass of another (same material) has only ∛8 = 2 times the diameter.
- Kepler’s third law. An orbit’s size is proportional to the cube root of the square of its period, so the semi-major axis in astronomical units is ∛(T²) with T in years.
Cube roots and the cubic equation
Solving x³ = 54 gives exactly one real solution, x = ∛54. The equation also has two complex solutions, ∛54 multiplied by (−1 ± i√3)/2, which become important when solving general cubic equations. The cubic equation calculator finds all three. For square roots use the square root calculator, and to see which numbers are perfect cubes, browse the list of perfect cubes.
Frequently asked questions
What is a cube root?
The cube root of x is the number that, used as a factor three times, gives x. ∛27 = 3 because 3 × 3 × 3 = 27. It is the inverse of cubing, and it can be written as x^(1/3).
Can you take the cube root of a negative number?
Yes. Unlike square roots, cube roots of negative numbers are real: ∛−27 = −3 because (−3)³ = −27. Every real number has exactly one real cube root.
How do you simplify a cube root?
Factor the number and pull out every group of three equal factors. 54 = 2 × 3³, so ∛54 = 3∛2. A cube root is fully simplified when no perfect cube other than 1 divides the radicand.
How do I find a cube root without a calculator?
Bracket it between perfect cubes, then refine. ∛100 lies between 4 (4³ = 64) and 5 (5³ = 125). Newton's method, g → (2g + x/g²)/3, sharpens the estimate quickly: starting at 4.5 gives 4.646, close to the true 4.6416.