Cube Root

The cube root ³√a or 3rd root of a number b is such that b³ = a.

By definition, if ³√a is multiplied three times it gives b as result.

The term is usually denoted with the √ symbol and the index 3, but it can also be written in exponential form with the base a and the exponent 1/3 as explained further below on this page.

In this article you can learn everything about these numbers.

We also show you the properties along with examples of cube roots.

Make sure to check out our calculator, too.

Calculator

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Cube Root:

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In case you know exponentiation, then you can think of the cube root (cbrt) of a number as the inverse operation to elevating a number to the power of three.

Whereas in exponentiation elevating a number a to the power of three means a³ = b, the cbrt b is defined as b = a^1/3.

For example with a = 27 we get:

³√27 = (3³)^1/3 = 3^3/3 = 3¹ = 3.

In other words, the cube root of 27 is 3, because 3 times 3 times 3 is 27.

Likewise, the cube root of 8 = 2, and the cube root of 125 = 5.

In contrast to a square root, a cbrt ³√a has only one real value:

³√a =b

Symbol

The third root symbol is ³√x . This is called the radix sign with index 3.

In Microsoft Word you can use superscript to write the index of 3, along with the radical sign you can insert by means of the Insert –> Symbol menu.

More information about the root symbol can be found on our home page and in the next section.

Cube Root Parts

The parts of a cube root are as follows:

The radix sign indicates that it is a mathematical root, and the index of three tells us that it is the 3rd root.

The number below the radix is called the radicand.

The result of the mathematical operation is denoted by the equal sign and called the 3rd root.

How to Find the Cube Root

The easiest way to find the cube root of a number is using a calculator like the one you can find in the first paragraph.

In the absence of a calculator we recommend to use the guess and check method:

1. Find the two perfect cubes your number is between. The cbrt of your number must be between the roots of these perfect cubes.
2. For example, to find ³√47 proceed as follows: 47 lies between the perfect cube of 3, 27, and the perfect cube of 4, 64. Therefore, ³√47 must be between ³√27 and ³√64, that is between 3 and 4.
3. Build the sum of these two roots to obtain 7, and divide the result by 2 to get 3.5. Then raise it to the power of 3: (3.5)³ = 42.875. The result is less than 47, so ³√47 must be bigger than 3.5.
4. Build the sum of 3.5 and 4, divide it by 2 and ^3 3.75 to obtain 52.734375. This is more than 47, so ³√47 must be less than 3.75.
5. Next, build the sum of 3.5 and 3.75 and divide it by 2. Then elevate 3.625 to the power of 3 to obtain 47.634765625, a bit more than 47. Thus, ³√47 must be a bit less than 3.625.
6. Proceeding in the same way until you are close enough to 3.6088… = ³√47

We guess this answered the question how to find cube root manually.

For methods of computing these numbers please follow the references link at the end of this page.

About our Cube Root Calculator

To use our calculator at the top of this page enter any real number, the calculation is done automagically.

To compute another number hit the reset button first.

If our tool has been helpful to you then bookmark it now as cube root calculator.

This table is a shortcut to the most searched items in Google:

Properties

With and , the properties are as follows:

• = -a if a < 0 and a if ≥ 0

Read on to see the properties in use:

Examples

We use the list of properties above to show you some examples in the order of appearance:

Frequently searched terms on this site include:

• Cube Root of 216
• Cube Root of 125
• Cube Root of 4096

Cube Root of Negative Number

The cube root of a negative number is a negative number because the multiplication of any negative number with itself three times is negative.

As opposed to square roots, every number in has exactly one corresponding cbrt.

This can been seen easily by looking at the graph in the next section.

Cube Root Function

Last, but not least, here is the cube root function f(x) = ,


This function maps the set of real numbers onto their cube roots. In geometry, the function f(x) = ³√x maps the area of a cube to its side length.

FAQs

Click on the question which is of interest to you to see the collapsible content answer.

Does a Cube Root Have Two Answers?

All real numbers ≠ 0 have exactly one real cube root, and, in addition, a pair of complex conjugate cube roots. 0 only has one cube root: 0.

Is a Cube Root a Function?

In mathematics, a cube root is the concept of a number x such that ∛y = x; the corresponding function is: f(x) = ∛y.

How Do You Write the Third Root?

The third root can be written as ∛x or x^(1/3).

What are Real Cube Roots?

Real cube roots belong to the set of real numbers denoted using the symbol R.

What Does Cube Mean?

In mathematics, a cube is the concept of a number y such that x³ = y.

What Does Cube Root Mean?

In mathematics, a cube root is the concept of a number x such that ∛y = x.

Table

Bottom Line

This ends our article about cbrt. In the search form in the sidebar you can find many cubes roots we have already calculated for you.

At the same place you can also look for square roots, cubes, squares as well as perfect squares.

If you are pleased with our explanations then please press the like button below to let the world know about our website.

For comments and questions use the form below.

We try to answer all questions quickly, yet, for time constraints we currently don’t provide tuition. More Information, particularly about complex numbers:

• Cube Root on Wikipedia

– Article written by Mark, last updated on August 2nd, 2023

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