# Cubed

When a value is followed by the term "cubed," it means that the value is being raised to the power of three. For example, three cubed is written as follows:

In the above figure, "cubed" refers to the exponent, 3.

Cubing a value (raising it to the power of 3) just means to multiply the number by itself two times: 3^{3} = 3 × 3 × 3 = 27. One way to visualize this is to use a cube that is made up of cubes with side lengths of 1 unit (unit cube).

Each of the 6 faces of the cube above has side lengths of 3 units. The total number of unit cubes that make up the whole cube with side lengths of 3 units is 3 x 3 x 3, or 3^{3} = 27. This is one way to view the concept of volume (as the number of unit cubes that can fit in the space).

## Perfect cubes

Perfect cubes are the cubes of the integers. Memorizing or at least recognizing perfect cubes can be useful, particularly when solving algebraic equations. Below is a table showing the perfect cubes from 0-12.

Integer | Squares |
---|---|

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 |

## Cubes of negative numbers

The cube of a negative number is always negative. This is because of the nature of negative numbers. A negative value multiplied by another negative is positive. A negative value multiplied by a positive one is negative. In other words, if there are an odd number of negative values being multiplied, the result will be negative. Since a cube involves multiplying the same three numbers, the cube of a negative number will always result in a negative:

(-3)^{3} = (-3) × (-3) × (-3) = 9 × -3 = -27