Cube Root Calculator

This calculator can be used for finding all cube roots of the given number. It finds both real and imaginary roots.

Directions for use

To find the cube root of a number, enter that number into the input field and press "Calculate." The calculator will demonstrate the answer in two parts: the "principal (real) root", and "all roots," where "all roots" include the principal root and the imaginary roots.

The calculator accepts positive and negative integers as inputs. Fractions and imaginary numbers are not accepted. Note that if you use a fraction or an imaginary number as an input, this cube roots calculator will automatically disregard everything following the first non-number symbol. For example, if you enter 8/15, the calculator will calculate the cube root of 8; if you enter 5 + 3i, the cube root of 5 will be calculated.

Cube root definition

The cube root of a number is defined as the number that has to be multiplied three times to get the original number. The cube root of x is commonly denoted as ∛x. According to the definition, y is the cube root of x:

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

if

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

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

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

Cube root operation is the reverse of finding the cube operation. To find the cube of a number, that number has to be multiplied 3 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, the cube root of which is an integer. For example, 8 is a perfect cube since:

$$\sqrt[3]{8}=\sqrt[3]{2×2×2}=2$$

Since integers are whole numbers that can be positive and negative, perfect cubes can be both positive and negative. For example, -8 is a perfect cube since:

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

0 is also an integer and

$$\sqrt[3]{0}=\sqrt[3]{0×0×0}=0$$

Therefore, 0 is also a perfect cube.

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

∛4 ≈ 1.58740105

which is not an integer.

Cube root properties

A cube root of a negative number is defined as the negative of the cube root of a positive number, i.e.,

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

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, use the prime factorization method:

1. Find prime factors of the number.
2. Divide prime factors into groups containing three factors that are the same.
3. Take one factor of each of the groups, and multiply them to get the final answer.

For example, let's find all the real cube roots of 3375, ∛3375:

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

Therefore, ∛3375 = 15.

If the prime factors of a number do not form groups of three, the number is not a perfect cube, and we cannot use this method to find the cube root.

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

If the given number is greater than -1 and less than 1, it cannot be a perfect cube since by definition, a perfect cube is a number, the cube root of which is an integer. Any number y from the interval -1 < y < 1 that is not 0 cannot be a perfect cube. However, sometimes finding the real cube root of such a number can be relatively easy.

For example, let's find all the real cube roots of -0.000125. This number is not an integer. Therefore, we cannot use the prime factorization method described above.

But we can easily notice that -0.000125 = -125 × 10⁻⁶. Therefore,

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

Applying the multiplication property of the cube root, 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:

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

It is easy to notice 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, 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 are used in real life to find the side length of any cubic object. For example, if you know a box's volume and want to find how high it is, check whether it would fit somewhere. Or, if you need to estimate the amount of paint, you would need to paint the walls of a cubic room. Or, if you need to count the number of tiles, you need to cover the floor of a cubic room with a known volume.

The cubic volume of wood

Imagine building a house and finding an ad for 64 cubic meters of wood for sale. What would the dimensions of that volume of wood be in length, width, and height?

To solve this problem, you must find the cube root of 64. The length of the side of the imaginary cube that would help you describe this volume would be ∛64 = 4. Thus, from the original data on the cubic volume of wood, we have a different idea of the size of such a volume.