Cube root


The cube root is a mathematical operation that, from a positive real number, returns another positive real number which, raised to three, results in the initial number.

In other words, given a positive real number, the cube root finds another positive real number by which multiplied by itself three times results in the given number.

Beyond the cube root

The difference between a cube, square, and higher degree root is the small number that appears at the beginning of the root, n, and indicates the degree of the root. This number is called the index.

Scheme of a root

Due to the great use of the square root, when we use a cube root we must indicate it with the index as follows:


This avoids confusion and errors in the notation.

The roots and the coins

In the same way that coins have heads and tails, roots also have two sides:

The face would be the best known side:


The cross would be the least known side:


Although they seem different at first glance, like the heads and tails of a coin, they are equivalent since both express a root but one contains a power (tails) and the other a radicand (heads).

To understand that both expressions represent the same content, we will draw two ways to represent the cube root. Taking into account that both equations are equivalent, their functions will be superimposed and only one of the two will be seen. To avoid this overlap, we will add a negative sign to the power in order to differentiate them and see their symmetry.


You can try to represent both the expression that carries the radicand and the expression that carries the power and you will see that the functions coincide. So, we can express a root of the two ways. The most common way to express a root is with the radicand, but we can also express a root using the power.

Cube root examples

Calculation and result of some roots:


As can be seen, the results are positive real numbers. We are used to finding natural roots, but we can also find roots with decimals.

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