Cube root calculator
Enter a number and a root index (3 for the cube root). The result is the real root, so the cube root of −8 is −2. For whole numbers the tool also tells you whether it is a perfect power and writes the root in simplest radical form.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Type the number. Negatives and decimals are fine.
- Leave the index at 3 for the cube root, or enter another whole number from 2 to 100.
- Read the root. For whole numbers, check the perfect-power line and the simplified form.
How it works
The cube root uses the exact built-in cube root function and other roots use a power of 1/n, snapped to a whole number when that whole number raised to n gives back the input exactly. The perfect-power test and the simplification work on exact integers: the tool divides out p^n for p = 2, 3, 4 … while it still fits, collecting the factors p on the outside of the radical.
Worked example
∛54:
- 54 = 27 × 2 = 3³ × 2, so it is not a perfect cube.
- Simplified form: 3∛2.
- Decimal: 3.77976314968, and it lies between the perfect cubes 27 (3³) and 64 (4³).
And ∛−8 = −2, a perfect cube.
Tips
- To check a result, cube it: the “Root raised to the power n” row should return your number.
- Decimals like 0.027 work (∛0.027 = 0.3) but skip the radical form, which only applies to whole numbers.
- Memorize the cubes of 1 to 10 to spot perfect cubes quickly.
Frequently asked questions
What is the cube root of a number?
The cube root of x is the number that gives x when multiplied by itself three times. Since 3 × 3 × 3 = 27, the cube root of 27 is 3, written ∛27 = 3.
Can you take the cube root of a negative number?
Yes. Unlike square roots, cube roots of negatives are real, because a negative times a negative times a negative is negative. ∛−8 = −2 since (−2)³ = −8. The same holds for every odd index, such as the 5th root.
What is a perfect cube?
A perfect cube is a whole number that is the cube of another whole number: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Their cube roots are whole numbers.
How do I simplify a cube root?
Pull out the largest perfect cube factor. 54 = 27 × 2, so ∛54 = ∛27 × ∛2 = 3∛2. The calculator does this for whole numbers up to 1,000,000,000,000 and any index.
Why does an even root of a negative number give an error?
No real number multiplied by itself an even number of times gives a negative. Those roots exist only as complex numbers, which this tool does not handle.