Cube Root Calculator
Find the cube root of any positive or negative number with step-by-step work.
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How to Find the Cube Root
The cube root of a number x is the value that, when cubed (multiplied by itself three times), equals x. Written ∛x or x^(1/3). The cube root of 27 is 3 because 3³ = 27. Unlike square roots, cube roots of negative numbers are real: ∛(−8) = −2.
Perfect Cubes
Perfect cubes are numbers whose cube root is an integer: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, etc. For non-perfect cubes, the cube root is irrational.
Cube Roots of Negative Numbers
The cube root of a negative number is always negative: ∛(−27) = −3 because (−3)³ = −27. This differs from square roots, where negative inputs have no real solution. Cube roots are defined for all real numbers.
Simplifying Cube Roots
To simplify ∛n: find the largest perfect cube factor of n. For example, ∛54 = ∛(27 × 2) = ∛27 × ∛2 = 3∛2.
Applications
Cube roots appear in volume calculations (side of a cube given its volume), physics (density and dimensional analysis), and engineering. If a cube has volume 125 cm³, its side length is ∛125 = 5 cm.
Worked Example: Simplifying ∛250
Step 1 — Find the prime factorization of 250: 250 = 2 × 125 = 2 × 5³.
Step 2 — Identify the perfect cube factor: 5³ = 125 is a perfect cube. So 250 = 125 × 2.
Step 3 — Simplify: ∛250 = ∛(125 × 2) = ∛125 × ∛2 = 5∛2.
Step 4 — Decimal check: 5∛2 ≈ 5 × 1.2599 ≈ 6.299. And indeed ∛250 ≈ 6.299.
Perfect Cubes Reference Table
| n | n³ | ∛(n³) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 8 | 2 |
| 3 | 27 | 3 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 6 | 216 | 6 |
| 10 | 1,000 | 10 |
Frequently Asked Questions
∛8 = 2, because 2³ = 2 × 2 × 2 = 8.
Yes. ∛(−27) = −3, because (−3)³ = −27. Cube roots of negative numbers are real and negative.
Square root (√x) asks: what number squared equals x? Cube root (∛x) asks: what number cubed equals x? Square roots of negatives are imaginary; cube roots of negatives are real.
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 (these are 1³ through 10³).
If a cube has volume V, its side length is ∛V. For example, volume 216 cm³ → side = ∛216 = 6 cm.
Find the two perfect cubes your number falls between. For ∛100: since 4³=64 and 5³=125, the answer is between 4 and 5. Since 100 is closer to 125 than to 64, ∛100 ≈ 4.6. The precise value is ≈ 4.6416.
The cube root of a fraction equals the cube root of the numerator divided by the cube root of the denominator: ∛(1/8) = ∛1 / ∛8 = 1/2 = 0.5. This generalises to ∛(a/b) = ∛a / ∛b for any positive values.