# Cubes and Cube Roots

*To understand cube roots, first we must understand cubes ...*

## How to Cube A Number

To **cube** a number, just use it in a multiplication **3 times**...

### Example: What is 3 Cubed?

3 Cubed | = | ||

= | 3 × 3 × 3 | = 27 |

Note: we write "3 Cubed" as 3^{3}

(the little ^{3} means the number appears three times in multiplying)

## Cubes From 0^{3} to 6^{3}

0 cubed | = | 0^{3} | = | 0 × 0 × 0 | = | 0 |

1 cubed | = | 1^{3} | = | 1 × 1 × 1 | = | 1 |

2 cubed | = | 2^{3} | = | 2 × 2 × 2 | = | 8 |

3 cubed | = | 3^{3} | = | 3 × 3 × 3 | = | 27 |

4 cubed | = | 4^{3} | = | 4 × 4 × 4 | = | 64 |

5 cubed | = | 5^{3} | = | 5 × 5 × 5 | = | 125 |

6 cubed | = | 6^{3} | = | 6 × 6 × 6 | = | 216 |

## Cube Root

A **cube root** goes the other direction:

3 cubed is 27, so the **cube root of 27 is 3**

3 | 27 |

The cube root of a number is ...

... a special value that when **cubed** gives the original number.

The cube root of **27** is ...

... **3**, because **when 3 is cubed** you get **27**.

Note: When you see "root" think
In this case the tree is "27", and the cube root is "3". |

Here are some more cubes and cube roots:

4 | 64 | |

5 | 125 | |

6 | 216 |

### Example: What is the Cube root of 125?

Well, we just happen to know that **125 = 5 × 5 × 5** (if you use 5 three times in a multiplication you will get 125) ...

**... so the cube root of 125 is 5**

## The Cube Root Symbol

This is the special symbol that means "cube root", it is the cube root. |

You can use it like this: (we say "the cube root of 27 equals 3")

## You Can Also Cube Negative Numbers

Have a look at this:

So the **cube root** of −125 is −5

## Perfect Cubes

The Perfect Cubes are the cubes of the whole numbers:

PerfectCubes | |

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 |

It is easy to work out the cube root of a perfect cube, but it is **really hard** to work out other cube roots.

### Example: what is the cube root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

- Let's try 3.5:
*3.5 × 3.5 × 3.5 = 42.875* - Let's try 3.2:
*3.2 × 3.2 × 3.2 = 32.768* - Let's try 3.1:
*3.1 × 3.1 × 3.1 = 29.791*

We are getting closer, but very slowly ... at this point, I get out my calculator and it says:

*3.1072325059538588668776624275224*...

... but the digits just go on and on, without any pattern. So even the calculator's answer is **only an approximation ! **

(Further reading: these kind of numbers are called surds which are a special type of irrational number)

(Further reading: these kind of numbers are called surds which are a special type of irrational number)