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.

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 is defined as a3 = b, the cbrt b is defined as b = a1/3. For example with a = 27 we get:

3√27 = (33)1/3 = 33/3 = 31 = 3.

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

In contrast to a square root, a cbrt 3√a has only one real value: \sqrt[3]{a}=b.

Cube Root Symbol

The cube root symbol is \sqrt[3]\. 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 cube root.
Cube Root

How to Find the Cube Root of a Number

The easiest way to find the cube root of a number is using a calculator like the one you can find in the next paragraph a little bit further down. In the absence of a calculator we recommend to use the guess and check method:

Find the two perfect cubes your number is between. The cbrt of your number must be between the roots of these perfect cubes.

For example, to find \sqrt[3]{47}\ proceed as follows: 47 lies between the perfect cube of 3, 27, and the perfect cube of 4, 64. Therefore, \sqrt[3]{47}\ must be between \sqrt[3]{27}\ and \sqrt[3]{64}\, that is between 3 and 4.

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 \sqrt[3]{47}\ must be bigger than 3.5.

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 \sqrt[3]{47}\ must be less than 3.75.

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, \sqrt[3]{47}\ must be a bit less than 3.625.

Proceeding in the same way until you are close enough to 3.6088... = \sqrt[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.

Cube Root Calculator

To use this cube root calculator enter any real number. Then press calculate. To compute another number hit the reset button first.


Number:
Cube Root:

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

Cube Root Property

With a,b\hspace{3px} \varepsilon\hspace{3px} \mathbb{R^{+}} and k,m \hspace{3px} \varepsilon\hspace{3px} \mathbb{N}, the properties of cube roots are as follows:

  • \sqrt[3]{a^{3}} = |a|\hspace{3px} = -a if a < 0 and a if ≥ 0
  • \sqrt[3]{ab}= \sqrt[3]{a} \sqrt[3]{b}
  • \sqrt[3]{a^{m}} = a^{\frac{m}{3}} \Rightarrow \sqrt[3]{a} = a^{\frac{1}{3}}
  • \sqrt[3]{a/b}= \frac{\sqrt[3]{a}}{\sqrt[3]{b}}
  • \sqrt[3]{0} = 0
  • \sqrt[3]{a^{-m}} = \frac{1}{\sqrt[3]{a^{m}}}
  • \sqrt[3]{a^{m}} = \sqrt[3k]{a^{km}}
  • \sqrt[3]{\sqrt[m]{a}} = \sqrt[3m]{a} = \sqrt[m]{\sqrt[3]{a}}

Read on to see the properties in use:

Cube Root Examples

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

  • \sqrt[3]{64} = 4 ,\hspace{15px}\sqrt[3]{-64} = - 4
  • \sqrt[3]{216} = \sqrt[3]{8} \sqrt[3]{27} = 2 x 3 = 6
  • \sqrt[3]{6^{3}} = 6^{\frac{3}{3}} = 6^{1} = 6
  • \sqrt[3]{8/27}= \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = 2/3
  • \sqrt[3]{3^{-3}} = \frac{1}{\sqrt[3]{3^{3}}} = \frac{1}{\sqrt[3]{27}} = 1/3
  • \sqrt[3]{7^{m}} = \sqrt[3k]{7^{km}}
  • \sqrt[3]{\sqrt[m]{125}} = \sqrt[3m]{125} = \sqrt[m]{\sqrt[3]{125}}

Cube Root of Negative Number

The cube root of a negative number is a negative number because every negative number multiplied three times with itself is negative. In contrast to square roots, every number in \mathbb{R} 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) = \sqrt[3]{x}, x \varepsilon \mathbb{R}
Cube Root Function
This function maps the set of real numbers onto their cube roots. In geometry, the function f(x) = \sqrt[3]{x}\hspace{3px} maps the area of a cube to its side length.

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.

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