Cube Root Calculator

This calculator can be used to easily find all the cube roots of any given number. It accurately computes both the real and imaginary roots, making it an essential tool for your math calculations.

Directions for use

To find the cube root of a number, simply enter that value into the input field and click "Calculate." The calculator will display the results in two distinct sections: the "principal (real) root" and "all roots." The "all roots" section includes both the principal real root and its corresponding imaginary roots.

This cube root calculator accepts both positive and negative integers or decimals as inputs. However, fractions and complex imaginary numbers are not supported. Please note that if you input a fraction or an imaginary number, the tool will automatically ignore everything following the first non-numeric symbol. For example, if you enter 8/15, the calculator will evaluate the cube root of 8. Similarly, if you enter 5 + 3i, it will calculate the cube root of 5.

Cube root definition

The cube root of a number is defined as the mathematical value that, when multiplied by itself three times, equals the original number. The cube root of x is commonly denoted by the symbol ∛x. By definition, y is the cube root of x:

$$y=\sqrt[3]{x}$$

if

$$y \times y \times y = x$$

Taking the cube root of a number, ∛x, is mathematically equivalent to raising that number to the power of 1/3:

$$\sqrt[3]{x}=x^{\frac{1}{3}}$$

The cube root operation is the exact inverse of finding the cube of a number. To find the cube of a number, you multiply it by itself three times:

$$y^3 = y \times y \times y = x$$

And inversely:

$$\sqrt[3]{x}=\sqrt[3]{y×y×y}=y$$

Perfect Cubes

A perfect cube is a number whose exact cube root is an integer. For example, 8 is a perfect cube because:

$$\sqrt[3]{8}=\sqrt[3]{2×2×2}=2$$

Because integers are whole numbers that can be either positive or negative, perfect cubes can also be both positive and negative. For instance, -8 is a perfect cube because:

$$\sqrt[3]{-8}=\sqrt[3]{-2×-2×-2}=-2$$

Zero (0) is also an integer, and:

$$\sqrt[3]{0}=\sqrt[3]{0×0×0}=0$$

Therefore, 0 is considered a perfect cube.

On the other hand, 4 is not a perfect cube because the principal real cube root of 4 is:

∛4 ≈ 1.58740105

which is a repeating decimal and not an integer.

Cube root properties

The cube root of a negative number is defined as the negative equivalent of the cube root of a positive number. In mathematical terms:

$$\sqrt[3]{-x}=-\sqrt[3]{x}$$

For example:

$$\sqrt[3]{-27}=-\sqrt[3]{27}=-3$$

Multiplication property of cube roots:

$$\sqrt[3]{x}×\sqrt[3]{y} =\sqrt[3]{x×y}$$

How to calculate cube root

Calculating the real cube root of a perfect cube

To find the cube root of a number by hand, the prime factorization method is highly effective:

1. Find the prime factors of the number.
2. Divide these prime factors into groups containing three identical factors.
3. Take one factor from each group, and multiply them together to get your final answer.

For example, let's find the real cube root of 3375 (∛3375):

1. Finding the prime factors of 3375, we get: 3375 = 3 × 3 × 3 × 5 × 5 × 5.
2. Dividing them into groups of three identical factors, we get: 3375 = (3 × 3 × 3) × (5 × 5 × 5).
3. Finally, taking one factor from each group and multiplying them, we get: 3 × 5 = 15.

Therefore, ∛3375 = 15.

If the prime factors of a number cannot be grouped into sets of three identical numbers, the number is not a perfect cube, and this method cannot be used to find an integer cube root.

Calculating the real cube root of a number greater than -1 and less than 1 (excluding 0)

If a given number falls strictly between -1 and 1, it cannot be a perfect cube. By definition, a perfect cube must yield an integer cube root, and any non-zero number y within the interval -1 < y < 1 will not. However, calculating the real cube root of such fractions and decimals can still be relatively straightforward.

For example, let's find the real cube root of -0.000125. Because this number is a decimal and not an integer, we cannot use the standard prime factorization method described above.

However, we can easily observe that -0.000125 is mathematically equivalent to -125 × 10⁻⁶. Therefore:

$$\sqrt[3]{-0.000125}=\sqrt[3]{(-125)×10⁻⁶}$$

Applying the multiplication property of cube roots, we get:

$$\sqrt[3]{-0.000125}=\sqrt[3]{(-125)×10⁻⁶}=\sqrt[3]{(-125)}×\sqrt[3]{10⁻⁶}$$

Rewriting the cube root of the negative number as the negative of the cube root of the positive number, we get:

$$\sqrt[3]{(-125)}×\sqrt[3]{10⁻⁶}=-\sqrt[3]{(125)}×\sqrt[3]{10⁻⁶}$$

We can easily spot that 125 = 5 × 5 × 5, and 10⁻⁶ = 10⁻² × 10⁻² × 10⁻². Therefore:

$$\sqrt[3]{(125)}=\sqrt[3]{(5×5×5)}=5$$

and

$$\sqrt[3]{(10⁻⁶)}=\sqrt[3]{(10⁻²)×(10⁻²)×(10⁻²)}=10⁻²$$

Finally, substituting these values back, we get:

$$\sqrt[3]{(-0.000125)}=\sqrt[3]{((-125) × 10⁻⁶)}=\sqrt[3]{(-125)}×\sqrt[3]{(10⁻⁶)}$$

$$\sqrt[3]{(-125)}×\sqrt[3]{(10⁻⁶)}=-\sqrt[3]{(125)}×\sqrt[3]{(10⁻⁶)}$$

$$-\sqrt[3]{(125)}×\sqrt[3]{(10⁻⁶)}=-\sqrt[3]{(5×5×5)}×\sqrt[3]{(10⁻²)×(10⁻²)×(10⁻²)}=(-5)×10⁻²=-0.05$$

Real life examples

Cube roots have highly practical applications in everyday life, particularly when you need to determine the exact side lengths of cubic objects. For example, if you know the total volume of a box and need to find its height to ensure it fits comfortably in a storage space, a cube root calculation is required. Similarly, estimating the exact amount of paint needed for the walls of a perfectly cubic room, or calculating the number of tiles required to cover the floor of a cubic room with a known volume, relies on finding the cube root.

The cubic volume of wood

Imagine you are building a house and come across an advertisement selling 64 cubic meters of wood. What would the actual dimensions of that volume of wood be in terms of length, width, and height?

To solve this problem and visualize the space, you need to find the cube root of 64. By calculating ∛64 = 4, you determine that the side length of an imaginary perfect cube representing this total volume is 4 meters. This simple cube root calculation transforms abstract volume data into concrete, easy-to-understand dimensions (4m × 4m × 4m), giving you a crystal-clear idea of the scale of your materials.