Cube Root Calculator (∛n)

∛n with perfect-cube detection, simplified surd form, and decimal expansion.

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter any real number (positive or negative).
  • Read decimal + simplified surd.

About this calculator

The cube root of n is the value x where x³ = n. Unlike square roots, cube roots accept negative inputs (∛(−8) = −2). Perfect cubes: 1, 8, 27, 64, 125, …. Simplified surd: ∛54 = ∛(27 × 2) = 3∛2. JavaScript provides Math.cbrt which handles negatives correctly and runs fast on hardware. Useful in geometry (volume of a cube of size x = x³), finance (annualized growth from a 3-period total), and music (frequency ratios).

Frequently asked

Why does cube root accept negatives?+
Odd functions: y = x³ is one-to-one over all reals. (−2)³ = −8, so ∛(−8) = −2. Square root only defined for non-negatives in real domain.
Perfect cube examples?+
8 (=2³), 27 (=3³), 64 (=4³), 125 (=5³), 216 (=6³), 343 (=7³), 512 (=8³), 729 (=9³), 1000 (=10³).
How is it computed?+
Math.cbrt uses Newton iteration internally. Hardware-supported on most modern CPUs.
Surd simplification?+
Find the largest k where k³ divides n. ∛54 = ∛(27×2) = 3∛2.
When is cube root used?+
Volume → side: cube with volume V has side ∛V. Annualized growth from 3-period total: r = ∛(end/start) − 1.

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