CBSE Class 8CBSE
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Cubes - Some Interesting Patterns

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description

  • Adding consecutive odd numbers
  • Cubes and their prime factors

notes

i) Adding consecutive odd numbers: 
`1 = 1 = 1^3` 
`3 + 5 = 8 = 2^3`
`7 + 9 + 11 = 27 = 3^3`
`13 + 15 + 17 + 19 = 64 = 4^3`
`21 + 23 + 25 + 27 + 29 = 125 = 5^3`

ii) Cubes and their prime factors :
Consider the following prime factorisation of the numbers and their cubes. 

Prime factorisation of a number                      Prime factorisation of its cube
4 = 2 × 2                                    `4^3 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^3 × 2^3` 
6 = 2 × 3                                           `6^3 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2^3 × 3^3`
15 = 3 × 5                                    `15^3 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3^3 × 5^3`
12 = 2 × 2 × 3                              `12^3 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 
2^3 × 2^3 × 3^3 `

Do you remember that `a^m × b^m = (a × b)^m` . 
Each prime factor of a number appears three times in the prime factorisation of its cube.

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