Cube Root Calculator
This cube root calculator finds ∛x for any real number and shows how to get there. Example: ∛54 = 3.779763, which simplifies to 3 · ∛2 because 54 = 27 × 2.
Example: Number 54 → Cube root: ∛54 = 3.779763. Source: Wolfram MathWorld – Cube Root (real cube root, also for negative numbers). Updated: .
How it is calculated
The cube root of a number x is the number that gives x when multiplied by itself three times: ∛x = y means y³ = x. It is the same as raising x to the power 1/3. Because a negative number cubed stays negative, every real number has exactly one real cube root – ∛(−216) = −6.
How to find a cube root by hand
Split the number into prime factors and look for groups of three equal factors. Each complete group comes out of the root as a single factor; whatever is left stays under the root sign. If nothing is left, the number is a perfect cube.
Worked example
54 = 2 × 3 × 3 × 3 = 2 × 3³. The group 3³ leaves the root as 3, the 2 stays inside: ∛54 = 3 · ∛2. With ∛2 = 1.259921 that is 3.779763. A quick plausibility check: 3³ = 27 and 4³ = 64, so the answer must lie between 3 and 4.
More examples: ∛27 = 3 and ∛1,000 = 10 are perfect cubes. ∛16 = 2 · ∛2 = 2.519842, because 16 = 2³ × 2. ∛10 = 2.154435 cannot be simplified. Decimals work as well: ∛0.125 = 0.5.
Useful rules
- ∛(a × b) = ∛a × ∛b and ∛(a ÷ b) = ∛a ÷ ∛b.
- ∛(a³) = a for every real a, positive or negative.
- The first perfect cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000.
- A cube with volume V has an edge length of ∛V – the most common everyday use.
Frequently asked questions
What is the cube root of 54?
∛54 = 3.779763. In simplified form it is 3 · ∛2, since 54 = 3³ × 2.
Can you take the cube root of a negative number?
Yes. A negative number has a negative cube root: ∛(−216) = −6, because (−6)³ = −216. That is different from square roots, which have no real value for negative numbers.
How do I simplify a cube root?
Factor the number into primes and pull out every group of three equal factors. 16 = 2³ × 2, so ∛16 = 2 · ∛2 = 2.519842.
How do I know whether a number is a perfect cube?
Its prime factors all appear in multiples of three. 216 = 2³ × 3³ is a perfect cube with root 6; 54 = 2 × 3³ is not, because the factor 2 appears only once.
How do I type a cube root on a calculator?
Raise the number to the power 1/3, for example 10^(1/3) = 2.154435. Many pocket calculators also have a ∛ or x^(1/y) key.
Sources and legal basis
- Wolfram MathWorld – Cube Root (real cube root, also for negative numbers)
- Wolfram MathWorld – Cubic Number (Kubikzahlen)
- DLMF §4.2: NIST Digital Library of Mathematical Functions – Powers and roots
As of: · Editorial policy
Related tools
- Square Root CalculatorSquare root calculator with steps: square, cube and nth roots, simplified radical form (√72 = 6√2), prime factorisation and the Babylonian method.
- Exponent calculator with stepsCalculate powers, nth roots and logarithms with any base – step-by-step, with negative and fractional exponents, ln, log and scientific notation.
- Prime Factorization CalculatorPrime factorization calculator with steps: factor any number up to one trillion into primes, see the factor tree, test for primality and count divisors.
- Pythagorean Theorem CalculatorPythagorean theorem calculator: enter two sides to get the third side, both angles, area and height – with exact square roots, working and a right-angle check.
- Percentage CalculatorFree percentage calculator: find X% of a number, what percent one number is of another, percentage change, increase and discount – with formula and steps.
- Online CalculatorCalculator online, free and instant: type or tap and see the answer as you go – order of operations, brackets, percent, square root, steps and a history list.
- Square numbers and cube numbersSquare numbers as a list up to 200 with differences, plus cube, triangular and other figurate numbers. Check whether a number belongs, with root and neighbours.
- Surface area calculatorSurface area calculator for cube, cuboid, cylinder, cone, sphere, hemisphere, pyramid and torus: with lateral area, volume, litres, formula and steps.
- Linear Interpolation CalculatorLinear interpolation calculator: find the y value at an x between two known points, with the line equation, the inverse for x and the steps. Extrapolation too.