Calculate the cube root (∛x) of any number instantly. Get step-by-step solutions, perfect cube detection, and support for negative and decimal numbers.
A shipping container has a volume of 216 cubic feet. What is the length of one side?
Formula: ∛216 = 2161/3
Result: ∛216 = 6 feet
Since 6 × 6 × 6 = 216, each side of the container is 6 feet long.
Find the cube root of 64.
Calculation: 64 = 4 × 4 × 4
Result: ∛64 = 4
64 is a perfect cube (43 = 64). The cube root is a whole number.
Calculate the cube root of -125.
Calculation: -125 = (-5) × (-5) × (-5)
Result: ∛(-125) = -5
Unlike square roots, cube roots of negative numbers are negative. (-5)3 = -125.
The cube root of a number x is a value that, when multiplied by itself three times (cubed), equals x. It is denoted as ∛x or x1/3. Every real number has exactly one real cube root.
| Number (x) | Cube Root (∛x) | Calculation |
|---|---|---|
| 1 | 1 | 1 × 1 × 1 |
| 8 | 2 | 2 × 2 × 2 |
| 27 | 3 | 3 × 3 × 3 |
| 64 | 4 | 4 × 4 × 4 |
| 125 | 5 | 5 × 5 × 5 |
| 216 | 6 | 6 × 6 × 6 |
| 343 | 7 | 7 × 7 × 7 |
| 512 | 8 | 8 × 8 × 8 |
| 729 | 9 | 9 × 9 × 9 |
| 1000 | 10 | 10 × 10 × 10 |
Every real number has exactly one real cube root. Unlike square roots, there is no ambiguity with the sign.
The cube root of a negative number is negative. This is because cubing a negative number produces a negative result.
∛(a × b) = ∛a × ∛b. You can take the cube root of a product by multiplying the cube roots of each factor.
Cube roots are used in geometry (side length from volume), physics, engineering, and data analysis.
A cube root of a number x is a number y such that y × y × y = y³ = x. In other words, it is the inverse operation of cubing a number. The cube root is denoted with the radical symbol ∛ (called the cube root sign) as ∛x, or in exponential notation as x1/3.
For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8. Similarly, the cube root of -27 is -3 because (-3) × (-3) × (-3) = -27. Unlike square roots, cube roots of negative numbers are defined and negative — there is no need for imaginary numbers when dealing with real cube roots.
While square roots only produce non-negative results for real numbers, cube roots preserve the sign of the original number. This makes cube roots more versatile for many real-world applications, particularly in physics and engineering where negative values have physical meaning. Additionally, every real number has exactly one real cube root, whereas positive numbers have two square roots (positive and negative).
A perfect cube is a number that can be expressed as the cube of an integer. Common perfect cubes include 1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³), 216 (6³), 343 (7³), 512 (8³), 729 (9³), and 1000 (10³). If a number is a perfect cube, its cube root is an integer, making the calculation straightforward.
Cube roots appear in many fields of mathematics, science, and engineering. Here are some common applications:
Given the volume of a cube, the side length is the cube root of the volume. This is the most common real-world application of cube roots.
Cubic equations appear in fluid dynamics, thermodynamics, and structural engineering. Cube roots are used to solve for key variables in these equations.
The cube root transformation is sometimes used in statistics to normalize data distributions or reduce the impact of outliers.
Cube roots appear in certain financial models and economic growth calculations where cubic relationships exist between variables.