Cube Root Calculator for Any Real Number
Find the cube root of positive and negative numbers with a clear explanation
| Step | Value |
|---|
Cube root calculator results show whole numbers when possible
Cube root calculator results show whole numbers when possible and decimal approximations for non perfect cubes. The table below lists the cube roots of the first ten perfect cubes for quick reference.
| x | ∛x | Check (y³) |
|---|---|---|
| 1 | 1 | 1³ = 1 |
| 8 | 2 | 2³ = 8 |
| 27 | 3 | 3³ = 27 |
| 64 | 4 | 4³ = 64 |
| 125 | 5 | 5³ = 125 |
| 216 | 6 | 6³ = 216 |
| 343 | 7 | 7³ = 343 |
| 512 | 8 | 8³ = 512 |
| 729 | 9 | 9³ = 729 |
| 1000 | 10 | 10³ = 1000 |
Cube root of a number reverses the cubing operation
Cube root of a number reverses the cubing operation, finding a value y such that y multiplied by itself three times equals x. The cube root symbol uses a small 3 in the radical sign, written as the radical of x with index 3.
Cube roots as fractional exponents
Cube roots as fractional exponents show that the cube root of x equals x raised to the power of one third. This connection allows cube roots to follow the same rules as other exponents, including multiplication, division and powers of powers.
Finding the cube root of negative numbers
Finding the cube root of negative numbers works differently than square roots because cubing a negative number gives a negative result. This means every negative number has a real cube root, and that cube root is also negative. For example, the cube root of negative 64 equals negative 4.
Perfect cubes versus irrational cube roots
Perfect cubes versus irrational cube roots describes the two possible outcomes when taking a cube root. If the original number is a perfect cube like 8, 27 or 64, the cube root is a whole number. If the original number is not a perfect cube, like 10 or 20, the cube root is an irrational decimal that never terminates or repeats.
Cube roots compare differently to square roots for negatives
Cube roots compare differently to square roots for negatives because the index of the root, either 2 or 3, determines whether negative inputs are allowed. Square roots of negative numbers are not real numbers, but cube roots of negative numbers are always real because an odd number of negative factors produces a negative result.
Practical uses of cube roots
Practical uses of cube roots appear in geometry when finding the side length of a cube from its volume, in physics when relating quantities that scale with volume, and in finance when calculating compound growth rates over multiple periods.
Frequently asked questions about cube root calculations
A cube root of a number x is a value y such that y cubed equals x, meaning y times y times y equals x. For example, the cube root of 27 is 3 because 3 times 3 times 3 equals 27.
Yes, every negative number has a real cube root, and that cube root is also negative. For example, the cube root of negative 8 is negative 2 because negative 2 times negative 2 times negative 2 equals negative 8.
The cube root of x is the same as x raised to the one third power, written as x to the power of 1 over 3. This means cube roots can be calculated using fractional exponent rules.
A square root finds a value that when multiplied by itself twice equals the original number, and negative numbers have no real square root. A cube root finds a value that when multiplied by itself three times equals the original number, and every real number, including negatives, has exactly one real cube root.
For numbers that are not perfect cubes, the cube root is an irrational decimal number found using a calculator or numerical approximation methods. For example, the cube root of 10 is approximately 2.154435, which when cubed gives back approximately 10.
The cube roots of the first ten perfect cubes are whole numbers, where the cube root of 1 is 1, the cube root of 8 is 2, the cube root of 27 is 3, the cube root of 64 is 4, the cube root of 125 is 5, the cube root of 216 is 6, the cube root of 343 is 7, the cube root of 512 is 8, the cube root of 729 is 9, and the cube root of 1000 is 10.