How to Simplify a Cube Root
The cube root of a number n, written ∛n, is the value that produces n when multiplied by itself three times. Simplifying a cube root means rewriting ∛n as a·∛b, where a is a whole number pulled out of the radical and b is the smallest remaining "cube-free" number (no factor of b is a perfect cube). This calculator factors your radicand into primes, extracts every group of three matching factors, and reports both the simplified radical and its exact decimal value.
Formula and method
Break the radicand n into its prime factorization, n = p₁^e₁ · p₂^e₂ · ... . For each prime pᵢ with exponent eᵢ, every group of three copies can leave the radical: pᵢ contributes pᵢ^⌊eᵢ/3⌋ to the outside coefficient, and the leftover pᵢ^(eᵢ mod 3) stays under the root. Multiplying all the outside pieces together (and by any coefficient you entered) gives the final coefficient a, while multiplying the leftover pieces gives the simplified radicand b, so that the original expression a₀·∛n equals (a₀·a)·∛b. This follows directly from the product rule ∛(x·y) = ∛x·∛y and the identity ∛(a³) = a.
Worked example
Simplify ∛108. Factor 108 = 2² · 3³. The exponent on 3 is a full group of three, so 3 leaves the radical; the exponent on 2 is only 2, so 2² = 4 stays inside. That gives ∛108 = ∛(3³ · 4) = 3∛4 ≈ 4.7622. If the radicand is negative, factor its absolute value the same way and attach the negative sign to the outside coefficient, since ∛(-n) = -∛n.
Common mistakes
- Confusing cube roots with square roots: a cube root pulls out factors that appear three times, not two — ∛8 = 2 because 8 = 2³, not because 8 is "a perfect square."
- Dropping the sign: cube roots of negative numbers are real and negative (∛-27 = -3); do not treat a negative radicand as undefined the way you would for a square root.
- Stopping factorization early: if the remaining radicand still has a repeated prime factor appearing three or more times, it has not been fully simplified.
Real-world applications
- Algebra and precalculus coursework use simplified radical form to keep exact answers instead of rounded decimals.
- Geometry problems that involve cube volumes (V = s³) use cube roots to recover a side length from a volume, e.g. finding the edge of a cube-shaped tank.
- Engineering and physics formulas involving cubic scaling (such as period-radius relationships or volumetric scaling) often simplify more cleanly once cube roots are reduced.