Number
Unlike square roots, cube roots work for negative numbers: ∛-27 = -3.
Perfect Cubes Reference
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∛27
3
Perfect cube — exact integer
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.
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
Why does the cube root of a negative number exist when the square root does not?
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.
What are the perfect cubes?
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…).
Is cube root the same as n^(1/3)?
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.
Why doesn’t the cube root have a ± answer like square roots do?
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.
How is the cube root calculated?
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).
What is the cube root used for?
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.