Cube Root Calculator

Calculate the cube root of any real number instantly — negatives included (∛-27 = -3). Perfect cubes return exact integers, other roots get 9-significant-digit precision.

Cube Root

∛27

3

Perfect cube — exact integer

Verification3³ = 27
Also equals27^(1/3)
ClassificationPerfect cube

Number

Unlike square roots, cube roots work for negative numbers: ∛-27 = -3.

How the Calculation Works

The engine uses the real cube root function, which handles negatives natively (∛-27 = -3). Numerically it matches Newton's method applied to x³ − n = 0, iterating x → (2x + n/x²) / 3 — each step roughly doubles the correct digits, so convergence to machine precision takes only a few iterations. Perfect cubes are detected exactly and shown as integers (∛125 = 5); irrational roots are rounded to 9 significant digits, e.g. ∛10 = 2.15443469.

Perfect Cubes Reference

Quick Answer

The cube root of n is the number x where x × x × x = n. Unlike square roots, cube roots work for negative numbers: ∛-27 = -3 because (-3) × (-3) × (-3) = -27. Perfect cubes (1, 8, 27, 64, 125, …) return exact integers; other values return the root to 9 significant digits (∛10 = 2.15443469). Cube root is the same as raising to the power 1/3.

Key Facts

  • ∛n is the number x where x × x × x = n; cube root undoes cubing
  • Unlike square roots, cube roots are defined for negatives: ∛-27 = -3, ∛-8 = -2
  • Perfect cubes (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000) have exact integer roots
  • ∛n = n^(1/3) — the ⅓ power is the same operation
  • Every nonzero number has exactly one real cube root (no ± ambiguity like square roots)
  • ∛10 = 2.15443469, ∛2 = 1.25992105 — irrational, non-repeating decimals
  • ∛(a·b) = ∛a · ∛b for all real a, b (no sign restriction)
  • Odd-index roots (3rd, 5th, 7th, …) handle negatives; even-index roots (2nd, 4th, …) do not

Frequently Asked Questions

Signs cancel in pairs when multiplying. Squaring never produces a negative: (+x)² = x² and (-x)² = x², so √-n has no real answer. Cubing keeps the sign: (+x)³ = x³ and (-x)³ = -x³, so every negative number has a real cube root (∛-27 = -3). This generalizes: odd-indexed roots (3rd, 5th, 7th) accept negatives, even-indexed roots (2nd, 4th) do not.

Perfect cubes are integers whose cube roots are also integers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, … (n³ for n = 1, 2, 3, …), and their negatives -1, -8, -27, -64, … For example, 1728 = 12³. Any integer not on this list has an irrational cube root (∛2 = 1.25992105…).

Yes for real numbers: ∛n = n^(1/3). Beware in other software though — many programming languages compute negative powers via complex logarithms, so (-27)^(1/3) can return NaN or a complex number there. This calculator uses the real cube root function (Math.cbrt), which correctly returns -3 for ∛-27.

Cubing is a one-to-one function over the reals: it passes the horizontal line test, so each input has exactly one cube root. Squaring is not one-to-one (2² = (-2)² = 4), which is why square roots have two candidates and the √ symbol picks the principal one. For cube roots there is simply one real answer, with the same sign as the input.

This tool calls Math.cbrt, the hardware-backed real cube root. Numerically it is equivalent to Newton’s method applied to x³ - n = 0, iterating x → x - (x³ - n)/(3x²) = (2x + n/x²)/3, which converges quadratically from any nonzero start. Perfect cubes return exact integers (∛125 = 5); everything else is displayed to 9 significant digits (∛10 = 2.15443469).

Common uses: reversing a volume calculation (a cube of volume 1000 cm³ has edge ∛1000 = 10 cm), scaling laws in physics and engineering (the square-cube law), solving cubic equations in Depressed-cube form, and statistics such as the cube root of sample size for dimensional analysis.