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- Prime factorisation of a Number, Prime factorisation of its Square

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Consider the prime factorisation of the following numbers and their squares.

Prime factorisation of a Number |
Prime factorisation of its square |

6 = 2 × 3 | 36 = 2 × 2 × 3 × 3 |

8 = 2 × 2 × 2 | 64 = 2 × 2 × 2 × 2 × 2 × 2 |

12 = 2 × 2 × 3 | 144 = 2 × 2 × 2 × 2 × 3 ×3 |

15 = 3 × 5 | 225 = 3 × 3 × 5 × 5 |

You will find that each prime factor in the prime factorisation of the square of a number, occurs twice the number of times it occurs in the prime factorisation of the number itself.

Let us use this to find the square root of a given square number, say 324.

We know that the prime factorisation of 324 is

324 = 2 × 2 × 3 × 3 × 3 × 3

By pairing the prime factors, we get

324 =2 × 2 × 3 × 3 × 3 × 3

`= 2^2 × 3^2 × 3^2 = (2 × 3 × 3)^2`

So,`sqrt 324` =2 × 3 × 3 = 18