# Cube root

A cube root is a number that can be multiplied three times to give us the value under the radical symbol (also referred to as the radix). A cube root (or any root) is denoted as follows:

The only difference between the denotation of a cube root and any other root, referred to as an n^{th} root, is the index. The index changes to indicate the number of times the root is multiplied to equal the radicand. We can also think of the cube root as:

z^{3} = x | |

eg: | 2^{3} = 8 |

We can read the above example as "the cube root of 8 is 2."

Unlike square roots, it is possible to take the cube root of negative numbers. In general, it is possible to take the n^{th} root of a negative number as long as n is odd. If n is even, we can't take the root without using imaginary numbers since a negative number raised to an even exponent is positive. The cube root of a negative number is the same as the cube root of its positive value, just negative. For example, . On the other hand, -3 cubed is:

-3 × -3 × -3 = -27, so

## Finding cube roots

Cube roots are relatively simple if the radicand is a perfect cube. Like square roots, or any other radical, we usually try to simplify cube roots by manipulating the expression such that we are left with expressions involving a product of perfect cube(s) where possible. If this is not possible, we can only estimate the value of the cube root, though like all n^{th} roots, it is very difficult to estimate the roots of non-perfect cubes, and this is usually done using a calculator.

Example

Find :

It is helpful to remember some of the perfect cubes in order to be able to work with cube roots. Below is a table of the cubes of 0-20.

# | Perfect cube |
---|---|

0 | 0 |

1 | 1 |

2 | 8 |

3 | 27 |

4 | 64 |

5 | 125 |

6 | 216 |

7 | 343 |

8 | 512 |

9 | 729 |

10 | 1000 |

11 | 1331 |

12 | 1728 |

13 | 2197 |

14 | 2744 |

15 | 3375 |

16 | 4096 |

17 | 4913 |

18 | 5832 |

19 | 6859 |

20 | 8000 |