Cube Root Calculator
The cube root of a number is the value that, multiplied by itself three times, gives the number back: ∛27 = 3 because 3 × 3 × 3 = 27. Cube roots undo cubing the way square roots undo squaring, and they appear wherever volumes are involved, from finding a cube’s edge to engineering formulas.
This calculator returns the decimal value and the simplified radical form (∛54 = 3∛2), flags perfect cubes, and handles negative inputs correctly: unlike square roots, the cube root of a negative number is real and negative.
Cube Root Calculator
Decimal value, simplified radical form, and perfect-cube check.
| Simplified form | — |
| Check (cubed) | — |
| Type | — |
| All three roots | — |
Cube roots of negative numbers are real: ∛(−8) = −2. (Square roots of negatives are the ones that need complex numbers.)
How cube root simplification works
Take ∛54. Factor 54 into primes: 54 = 2 × 3³. For a cube root, group factors into triples: the triple of 3s comes out as a single 3, and the lone 2 stays inside. So ∛54 = 3∛2. The decimal, 3 × ∛2 ≈ 3.779763, is an approximation; the radical form is exact.
Worked examples
∛27 = 3 (perfect cube: 3³ = 27). ∛54 = 3∛2 (≈ 3.779763). ∛(−8) = −2, because (−2)³ = −8. ∛4 ≈ 1.587401, irrational, so no simpler radical form exists.
New: all three cube roots
Every nonzero number has three cube roots: one real and a complex pair. If r is the real root, the other two are -r/2 plus or minus i x r x sqrt(3)/2. The calculator now shows all three with a derivation note. For 27: 3, and -1.5 + 2.598076i and -1.5 - 2.598076i. Cubing any of them returns 27, which is how you can check the math.
What competitors miss
Calculator-online.net lists real and complex roots but gives no derivation, so students cannot see where the pair comes from. Omni Calculator teaches a clever mental trick that only works for perfect cubes. Cuemath shows steps but only the real root. This tool keeps the simplified radical form and prime-factor steps for the real root, and adds the full complex triple with the formula shown.
Worked example: the triple for 27
- Real root: 3.000000 (perfect cube, 3 x 3 x 3 = 27)
- Complex pair: -1.500000 + 2.598076i and -1.500000 - 2.598076i
- Check: |z|^3 = 27 and 3 x 120 degrees = 360 degrees, landing back on the real axis at 27
Quick cube root table
| Number | Cube root | Number | Cube root |
|---|---|---|---|
| 1 | 1 | 216 | 6 |
| 8 | 2 | 343 | 7 |
| 27 | 3 | 512 | 8 |
| 64 | 4 | 729 | 9 |
| 125 | 5 | 1000 | 10 |
Frequently asked questions
How many cube roots does a number have?
Three. Every nonzero real number has one real cube root and a complex conjugate pair; zero has a triple root of 0. The calculator shows all three.
Why does a negative number have a real cube root?
Because cubing preserves sign: (-2) x (-2) x (-2) = -8, so the cube root of -8 is -2. Square roots have no real answer for negatives; cube roots always do.
What is the simplified radical form?
Factoring out perfect cubes from inside the root. The cube root of 54 is 3 times the cube root of 2, because 54 = 27 x 2 and 27 is a perfect cube.
What is the cube root of a negative number?
A real, negative number. ∛(−8) = −2 because (−2) × (−2) × (−2) = −8. This works because three negatives multiply to a negative; with square roots the negatives pair up and vanish, which is why they need complex numbers.
How is a cube root different from a square root?
A square root asks “what times itself gives this number” (two factors); a cube root asks “what times itself three times gives this number.” Every real number has exactly one real cube root, including negatives and zero.
When is a cube root a whole number?
When the input is a perfect cube: 1, 8, 27, 64, 125, 216, and so on (n³ for integer n). The calculator flags these automatically.
Why show both the radical form and the decimal?
The radical form is exact and keeps algebraic work clean; the decimal is what you type into the next calculation or measurement. The “check” row cubes the decimal so you can see it lands back on your input.
Where are cube roots actually used?
Anywhere volume meets length: the edge of a cube with volume V is ∛V. They also appear in engineering formulas, compound-interest solving, and statistics (cube-root transformations of skewed data).