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Square Roots - Finding Square Root Through Prime Factorisation

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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

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