Let,

1 = 1 x 1 x 1;

8 = 2 x 2 x 2;

27 = 3 x 3 x 3;

125 = 5 x 5 x 5;

We observe that, each number is written as a product of 3 equal integers.

**We say that an integer N is a perfect cube if N can be written as a product of three equal integers. If N = m x m x m. we say N is the cube of m and write N = m ^{3} (read as cube of m or simply m-cube)**

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Consider a few more examples:

(-4) x (-4) x (-4) = -64 = (-4)^{3}

(-5) x (-5) x (-5) = -125 = (-5)^{3}

We see that the negative numbers are also perfect cubes.

Example 7: Find the cube of 6.

**Solution:**

6^{3} = 6 x 6 x 6 = 216

Example 8: What is the cube of 20?

**Solution:**

20^{3} = 20 x 20 x 20 = 400 x 20 = 8000.

Example 9: If a cube has side length 10cm, what is its volume?

**Solution:**

Volume, V = 10 x 10 x 10 = 1000m^{3}.

** Exercise 1.2.7**

- Looking at the pattern, fill in the gaps in the following:

2 | 3 | 4 | -5 | —— | 8 | —— |

2^{3 }= 8 |
3^{3} = —- |
—- = 64 | —- = —- | 6^{3}= —– |
—- = —- | — = -729 |

**Solution:**

2 | 3 | 4 | -5 | 6 | 8 | -9 |

2^{3 }= 8 |
3^{3} = 27 |
4^{3} = 64 |
(-5)^{3 }= -125 |
6^{3}= 216 |
8^{3} = 512 |
(-9)^{3} = -729 |

- Find the cubed of the first five odd natural numbers and the cubes of the first five even natural numbers. What can you say about parity of the odd cubes and even cubes?

**Solution:**

Odd cubes | Even cubes |

1^{3 }= 1 |
2^{3 }= 8 |

3^{3 }= 27 |
4^{3 }= 64 |

5^{3 }= 125 |
6^{3 }= 216 |

7^{3 }= 343 |
8^{3 }= 512 |

9^{3 }= 729 |
10^{3 }= 1000 |

Parity:

Cubes of odd numbers is always odd.

Cubes of even numbers is always even.

- How many perfect cubes you can find from 1 to 100 ? How many from – 100 to 100?

**Solution:**

We have 4 cubes from 1 to 100, those are,

1^{3 }= 1, 2^{3 }= 8, 3^{3 }= 27, 4^{3 }= 64. The next cube will be 5^{3 }= 125, which is larger than 100. so we have only 4 cubes from 1 to 100.

We have 8 cubes from -100 to 100. Those are, (-1)^{3 }= 1, (-2)^{3 }= 8, (-3)^{3 }= 27, (-4)^{3 }= 64 and 1^{3 }= 1, 2^{3 }= 8, 3^{3 }= 27, 4^{3 }= 64.

- How many perfect cubes are there from 1 to 500? How many are perfect square among cubes?

**Solution:**

We have 7 cubes from 1 to 500, i.e., 1^{3 }= 1, 2^{3 }= 8, 3^{3 }= 27, 4^{3 }= 64, 5^{3 }= 125, 6^{3 }= 216, 7^{3 }= 343.

- Find the cubes of 10, 30 , 100, 1000. What can you say about the zeros at the end?

**Solution:**

10^{3} = 1000

30^{3} = 27000

100^{3} = 1000000

1000^{3} = 10000000000

The number of zero of a cube are 3 times, the no. of zero of numbers

- What are the digits in the unit’s place of the cubes 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10? Is it possible to say that a number is not a perfect cube by looking at the digit in unit’s place of the given number, just like you did for squares?

**Solution:**

a | a^{3} |
digit in the units place |

1 | 1 | 1 |

2 | 8 | 8 |

3 | 27 | 7 |

4 | 64 | 4 |

5 | 125 | 5 |

6 | 216 | 6 |

7 | 343 | 3 |

8 | 512 | 2 |

9 | 729 | 9 |

10 | 1000 | 0 |

No, its not possible to tell a number is not a perfect cube by looking at the digit in unit’s place of the given number.

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