Cube Root Calculator

What is a cube root?

The cube root of a number x is the number y for which y × y × y = x. It is written ∛x, or as x^1/3. The name comes from geometry: a cube with side length y has volume y³, so the cube root of a volume is the side length of the cube that holds it.

∛x = y when y × y × y = xEquivalent to y = x^(1/3).

Numbers whose cube root is a whole number are called perfect cubes. 64 is one, because 4 × 4 × 4 = 64, so ∛64 = 4. Most numbers are not perfect cubes, and their cube roots are irrational decimals that never end, such as ∛2 = 1.2599...

Unlike a square root, a cube root is defined for every real number and has just one real value. In pure mathematics the cube root of −8 is −2. This calculator, however, returns nan for negative input. To handle a negative number, take the cube root of its positive counterpart and put the minus sign back.

How to use the cube root calculator

1. Type the number into the Number box. Use a dot for decimals, and a decimal for fractions (1/8 is 0.125).
2. Press Calculate cube root.

The answer is rounded to four decimal places. Whole numbers are shown bare (the cube root of 27 is 3), a result that needs one or two decimals shows two places (the cube root of 0.125 is 0.50), and larger numbers get thousands separators: the cube root of 1,000,000,000 is shown as 1,000. Results that round to zero at four places, such as the cube root of 0.000000000000001, appear as 0.

Perfect cubes and common cube roots

Knowing the first dozen or so perfect cubes lets you spot exact cube roots at a glance and bracket the rest.

Finding a cube root by hand

Start by bracketing the number between two consecutive perfect cubes. For 30, note that 27 < 30 < 64, so ∛30 lies between 3 and 4, and since 30 is only a little above 27 it is close to 3.

To sharpen the guess, use the Newton iteration, which averages your guess y with a correction based on x ÷ y².

new guess = (2 × y + x ÷ y²) ÷ 3

Example: ∛30 starting from y = 3. First step: (2 × 3 + 30 ÷ 9) ÷ 3 = (6 + 3.3333) ÷ 3 = 3.1111. Second step, carrying the unrounded 3.1111...: 3.107237. A third step gives 3.107233, so ∛30 ≈ 3.1072, which agrees with the calculator. Each step roughly doubles the number of correct digits.

Always verify by cubing: 3.1072 × 3.1072 × 3.1072 ≈ 29.999, which rounds to 30.

Properties of cube roots

∛(a × b) = ∛a × ∛b

∛(a ÷ b) = ∛a ÷ ∛b

∛(x³) = x

The product rule makes mental estimates easier. ∛216 = ∛(8 × 27) = 2 × 3 = 6, and ∛(1/8) = 1/2 because ∛1 = 1 and ∛8 = 2. Since the cube root is the exponent 1/3, a cube root of a sixth power is a square: ∛(x⁶) = x². For other roots see the root calculator, and for the exponent 1/2 use the square root calculator.

Where cube roots are used

• Cube dimensions. A cubic tank holding 500 cm³ has sides of ∛500 ≈ 7.9370 cm.
• Scaling. Volume grows with the cube of length, so doubling the volume of a shape multiplies each of its lengths by ∛2 ≈ 1.2599, an increase of about 26%.
• Spheres. The radius of a sphere from its volume V is ∛(3V ÷ 4π). For V = 1,000 that gives r ≈ 6.2035.
• Averages. The geometric mean of three numbers is the cube root of their product: the geometric mean of 2, 4 and 16 is ∛128 ≈ 5.0397. The geometric mean calculator handles any list.

Cube roots of fractions and decimals

The calculator takes decimals, so enter a fraction as a decimal number. To find the cube root of 27/64, type 0.421875 (27 ÷ 64) and get 0.75, since 3/4 cubed is 27/64. By the quotient rule you can also take the cube root of the numerator and the denominator separately: ∛27 = 3 and ∛64 = 4.

If a decimal is very long, the displayed cube root is rounded to four places. Cube the answer to see how close it comes to your starting number.