Cube Root Calculator

Use the cube root calculator to find the number that, multiplied by itself three times, gives your number. For example, ∛54 = 3∛2 ≈ 3.779763, because 54 = 27 × 2 and the cube root of 27 is 3. Negative numbers work too: ∛(−125) = −5, since (−5) × (−5) × (−5) = −125.

How to use the cube root calculator

1. Type any real number: a whole number, a decimal such as 0.125, a fraction such as 8/27, or a negative number.
2. Read the exact answer in simplest radical form (such as 3∛2) and the decimal. Use the Decimals menu for 4, 6, 10 or 20 places.
3. Open Show steps for the prime factorization, and Newton’s method to see how the decimal is found.
4. Use the nth root tab for square roots, fourth roots and higher, and the Cube tab to go the other way.

What is a cube root?

The cube root of x, written ∛x, is the number y with y × y × y = x. So ∛64 = 4 because 4³ = 64. In exponent form, ∛x = x^1/3. Geometrically, the cube root of a cube’s volume is the length of its edge: a cube of 343 cm³ has edges of ∛343 = 7 cm.

Unlike square roots, cube roots of negative numbers are real. Three negative factors give a negative product, so ∛(−8) = −2. Every real number has exactly one real cube root.

Perfect cubes from 1 to 20

A perfect cube is a whole number cubed. Knowing these makes cube roots of large numbers much easier to estimate.

The last digit of a perfect cube identifies the last digit of its root: cubes ending in 1, 4, 5, 6, 9 or 0 have roots ending in the same digit, while 2 and 8 swap, and 3 and 7 swap. For example, 5,832 ends in 2, so its root ends in 8, and it is 18³.

How to simplify a cube root

Write the number as a product of primes, then pull out every group of three equal factors.

1. Factor: 54 = 2 × 3 × 3 × 3 = 2 × 3³.
2. Each group of three equal factors comes out as one factor: 3³ comes out as 3.
3. Whatever is left stays inside: ∛54 = 3∛2.

More examples: ∛250 = ∛(5³ × 2) = 5∛2, ∛432 = ∛(2³ × 3³ × 2) = 6∛2, and ∛(−128) = −∛(4³ × 2) = −4∛2. If no prime appears three times, as in ∛10 = ∛(2 × 5), the root is already in simplest form. The prime factorization calculator helps with the first step.

Cube roots of fractions and decimals

Take the cube root of the top and the bottom separately: ∛(8/27) = ∛8/∛27 = 2/3. A decimal can be written as a fraction first: 0.125 = 1/8, so ∛0.125 = 1/2 = 0.5.

When the bottom is not a perfect cube, multiply inside the root to make it one. For ∛2.5 = ∛(5/2), multiply the top and bottom by 2² = 4: ∛(20/8) = ∛20/2 ≈ 1.357209.

How to estimate a cube root and Newton’s method

Start with the perfect cubes on either side. Because 27 < 54 < 64, the cube root of 54 is between 3 and 4, and closer to 4.

Newton’s method sharpens a guess quickly. For a cube root of a, repeat x → (2x + a/x²) ÷ 3:

After two rounds the answer is correct to six decimal places: ∛54 ≈ 3.779763. Each round roughly doubles the number of correct digits.

nth roots: square, fourth and higher roots

The nth root ⁿ√x is the number whose nth power is x. The same factoring idea works with groups of n equal factors: √72 = √(6² × 2) = 6√2 and ∜162 = ∜(3⁴ × 2) = 3∜2. Odd roots of negative numbers are real (⁵√(−243) = −3), but even roots of negative numbers are not real numbers, because an even power is never negative. For square roots, the square root simplifier and radical simplifier go further.

Where cube roots are used

• Volume to length. A cube-shaped tank that holds 1,000 liters (1 m³) has edges of ∛1 = 1 m; one that holds 8 m³ has edges of ∛8 = 2 m.
• Scaling. Doubling the volume of any shape multiplies each length by ∛2 ≈ 1.26, not by 2. That is why a box with twice the capacity looks only about a quarter wider.
• Spheres. The radius of a ball with volume V is ∛(3V ÷ 4π), so a ball of volume 33.5 cm³ has a radius of about 2 cm.

Common cube root mistakes

• Dividing by 3 instead of taking the cube root: ∛27 is 3, not 9.
• Saying a negative number has no cube root. ∛(−27) = −3 is a real number.
• Pulling out pairs instead of triples: ∛72 = ∛(2³ × 9) = 2∛9, not 6∛2.
• Rounding too early. Keep the exact form 3∛2 until the end, then round once.

Cube Roots practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

Can you take the cube root of a negative number?

Yes. The cube root of a negative number is negative, because a negative number multiplied by itself three times stays negative. For example, ∛(−64) = −4.

What is the cube root of 2?

∛2 ≈ 1.259921. It is irrational, so its decimal never ends or repeats, and the exact answer is written ∛2.

Is the cube root the same as raising to the power 1/3?

Yes, ∛x = x^1/3, so 8^1/3 = 2. The exponent calculator handles other fractional exponents such as 8^2/3 = 4.

How many cube roots does a number have?

Every real number has exactly one real cube root. In the complex numbers, every nonzero number has three cube roots: the real one and two complex ones. This calculator gives the real root.

Related calculators

• Square root simplifier: simplify square roots such as √72
• Radical simplifier: simplify radical expressions
• Exponent calculator: powers and fractional exponents
• Prime factorization calculator: break a number into primes