How to Calculate Cube Roots: Formula, Estimation, and Examples

How to Calculate the Cube Root

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. It's the inverse operation of cubing. Cube roots appear in geometry (finding the side of a cube from its volume), physics, and engineering.

The Formula

∛x = x^(1/3)

For a cube with volume V, the side length is:

s = ∛V

Perfect Cube Roots

Step-by-Step Example

Find ∛512.

Method 1: Recognize 512 = 8³, so ∛512 = 8

Method 2: Use 512^(1/3) on a calculator: 8

Method 3 (estimation): Since 7³ = 343 and 8³ = 512, we know ∛512 is between 7 and 8. Testing 8: 8 × 8 × 8 = 512. ✓

Non-Perfect Cube Roots

For non-perfect cubes, use prime factorization or a calculator.

∛100: Between 4³ = 64 and 5³ = 125, so between 4 and 5. 4.6³ = 97.34, 4.65³ = 100.54, so ∛100 ≈ 4.64

Negative Cube Roots

Unlike square roots, cube roots of negative numbers are real: ∛(−27) = −3 because (−3)³ = −27

Use our cube root calculator for any value.