Cube Root Calculator

The cube root is the number that must be multiplied by itself three times to get a certain number. It is denoted by $\sqrt[3]{x}$ and the cube root of x is the value a which satisfies the equation $a^3 = x$. Enter any number in this tool to calculate its result.

It supports both positive and negative numbers as well as integers and decimals. For perfect cubes it will show the exact value while for other numbers it will show rounded decimal values. For integer numbers, it also shows the simplified root form, and it can also calculate cube roots or any nth root.

What is a cube root?

Cubing a number means multiplying the number by itself three times in succession: $4^3 = 4 \times 4 \times 4 = 64$. The cube root reverses this process: $4^3 = 64$. So, the cube root of 64 is 4, which is written as $\sqrt[3]{64} = 4$.

The small 3 inside the radical sign is called the index of the radical, and it indicates that this is a cube root rather than a square root. A square root $\sqrt{x}$ is the result of multiplying a number by itself once; a cube root is the result of multiplying a number by itself twice.

How do you calculate a cube root?

There are three practical methods for calculating cube roots, and this tool uses the precise arithmetic underlying these methods.

Estimating between whole cubes:

If you find two perfect cubes that bracket the input number, then the answer is between the cube roots of those numbers. For example: if given 50, the nearest perfect cubes are $27 = 3^3$ and $64 = 4^3$, so $\sqrt[3]{50}$ must be between 3 and 4 (approximately 3.68).

Using prime factorization:

Factor the number into its prime factors and identify pairs of three identical factors. For example if we consider 216, then $216 = 2^3 \times 3^3$, the result is $\sqrt[3]{216} = 2 \times 3 = 6$. The simplified radical form shown below is constructed exactly this way.

Consider as exponents:

The cube root is the same as raising to the 1/3 power. This identity is why you can use a cube root directly in formulas or spreadsheets, where it will be written as x^(1/3).

Cube roots of negative numbers

Every real number, including negative numbers, has exactly one real cube root. If you multiply a negative number by itself and then again by another negative number, you get back to being a negative number, so -3 times -3 times -3 is -27, and the cube root of -27 is therefore -3. The rule is simple: take the cube root of the absolute value of the number and keep the minus sign.

This is fundamentally different from square roots. The square root of a negative number is not a real number, while $\sqrt[3]{-64} = -4$ poses no problem at all. This can be verified by filling in the above formula with a negative value.

Perfect cubes

A perfect cube is a number that is the result of multiplying an integer by itself three times. Calculating devices will identify these numbers and show the exact root, not a decimal. Here are the first twelve examples:

If you read this table backwards, you can directly determine the cube root. It has the form $\sqrt[3]{343} = 7$, where $\sqrt[3]{1000} = 10$ applies. The cube roots of all numbers that are not contained in this list are irrational numbers and consist of infinite non-periodic decimals.

In simplified root form:

Even with whole numbers that are not perfect cubes, the cube root can be expressed in the form $a\sqrt[3]{b}$. There will be no cubic factors remaining in b. You extract the largest perfect cube that is divisible by this number.

Let's take 250 as an example. Since it is a $250 = 125 \times 2 = 5^3 \times 2$, the 5 will be taken out of the radical sign and the 2 will remain inside the radical sign. The result is $\sqrt[3]{250} = 5\sqrt[3]{2}$. Whenever a whole number is entered, this calculator will output that simplified form.

Here are a few more examples: $\sqrt[3]{54} = 3\sqrt[3]{2}$, $\sqrt[3]{24} = 2\sqrt[3]{3}$, and $\sqrt[3]{16} = 2\sqrt[3]{2}$. If no factors can be extracted, such as with $\sqrt[3]{7}$, then the number is already in its simplest form.

Cube and nth root mode.

If you switch to the "Cube" mode, then you can directly calculate the $x^3$th root. This is useful for checking your result since the cube root of a number raised to that same number will give back the original number. If you switch to the "Nth root" mode, then you can use any arbitrary exponent for the root. For example, fourth root $\sqrt[4]{x}$, fifth root $\sqrt[5]{x}$ and so on. The cubic root is just the n-th root with n = 3, while the square root is n = 2.

Zum Beispiel: $\sqrt[4]{81} = 3$. Dies ist die $3^4 = 81$-te Wurzel. $\sqrt[5]{32} = 2$. Dies ist die $2^5 = 32$-te Wurzel. Da gerade Wurzeln negativer Zahlen keine reelle Zahl sind, werden diese markiert.

Applications of cube root:

One of the simplest applications is geometry. If the side length of a cube with volume V $\sqrt[3]{V}$ then a tank that holds 1,000 units of volume has a side length of 10 units. In general, the cube root can be used to find the length of a side of a cube if a certain quantity such as volume or mass is known and varies in proportion to the cube of the length.

The cube root also appears in algebraic equations such as the cubic formula and is used to solve some types of differential equations. In science and engineering it is sometimes used for dimensions that have a scaling effect on volume or mass, where there is a relationship between linear size and volume.

Example: The cube root of 100.

100 is not a perfect cube number. As it lies between $64 = 4^3$ and $125 = 5^3$, its cube root will lie between 4 and 5. If more digits are retained, the result becomes $\sqrt[3]{100} \approx 4.6416$. 100 has no perfect cubic factors other than 1 (as shown by $100 = 2^2 \times 5^2$) and is already a simplified radical $\sqrt[3]{100}$. Cubing 4.6416 gives 100, which is the check result displayed in the calculator.