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.

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Cube Root Symbol

The third root symbol is \sqrt[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.
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:

  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 \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.
  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 \sqrt[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 \sqrt[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, \sqrt[3]{47} must be a bit less than 3.625.
  6. 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 calculator enter any real number, the calculation is done automagically. To compute another number hit the reset button first.

Calculate \sqrt[3]{x}


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

Cube Root Property

With a,b \in \mathbb{R^{+}} and k,m \in \mathbb{N}, the properties are as follows:

  • \sqrt[3]{a^{3}} = |a| = -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 examples in the order of appearance:

  • \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}}

Frequently searched terms on this site include:

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.

As opposed 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 \in \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} maps the area of a cube to its side length.

Frequently Asked Questions

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.

What Does the Cube Root Symbol Look Like?

The cube root symbol is ∛.

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 of Cubes and Cube Roots

NumberSquareSquare Root
000
111
281.2599210499
3271.4422495703
4641.587401052
51251.7099759467
62161.8171205928
73431.9129311828
85122
97292.0800838231
1010002.15443469
1113312.2239800906
1217282.2894284851
1321972.3513346877
1427442.4101422642
1533752.4662120743
1640962.5198420998
1749132.5712815907
1858322.6207413942
1968592.6684016487
2080002.7144176166
2192612.7589241764
22106482.8020393307
23121672.8438669799
24138242.8844991406
25156252.9240177382
26175762.9624960684
27196833
28219523.0365889719
29243893.0723168257
30270003.107232506
31297913.1413806524
32327683.1748021039
33359373.20753433
34393043.2396118013
35428753.2710663102
36466563.3019272489
37506533.3322218516
38548723.3619754068
39593193.391211443
40640003.4199518934
41689213.4482172404
42740883.4760266449
43795073.5033980604
44851843.5303483353
45911253.5568933045
46973363.583047871
471038233.6088260801
481105923.6342411857
491176493.65930571
501250003.6840314986
511326513.7084297693
521406083.7325111568
531488773.7562857542
541574643.7797631497
551663753.8029524608
561756163.8258623655
571851933.8485011313
581951123.8708766406
592053793.8929964159
602160003.9148676412
612269813.9364971831
622383283.9578916097
632500473.9790572079
642621444
652746254.0207257586
662874964.0412400206
673007634.0615481004
683144324.0816551019
693285094.1015659297
703430004.1212852998
713579114.1408177494
723732484.1601676461
733890174.1793391964
744052244.1983364538
754218754.2171633265
764389764.2358235843
774565334.2543208651
784745524.2726586817
794930394.290840427
805120004.3088693801
815314414.3267487109
825513684.3444814858
835717874.3620706715
845927044.3795191399
856141254.3968296722
866360564.4140049624
876585034.4310476217
886814724.4479601811
897049694.4647450956
907290004.4814047466
917535714.4979414453
927786884.5143574355
938043574.5306548961
948305844.5468359438
958573754.5629026354
968847364.5788569702
979126734.5947008922
989411924.6104362921
999702994.6260650092
10010000004.6415888336

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.

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