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Find the square root of [tex]$1008$[/tex] by the prime factorization method.

Sagot :

Sure, let's find the square root of 1008 using prime factorization step-by-step:

1. Prime Factorization:
First, we need to break down 1008 into its prime factors.

- Start with the smallest prime number, 2:
[tex]\[ 1008 \div 2 = 504 \][/tex]
[tex]\[ 504 \div 2 = 252 \][/tex]
[tex]\[ 252 \div 2 = 126 \][/tex]
[tex]\[ 126 \div 2 = 63 \][/tex]

- Now move to the next smallest prime number, 3:
[tex]\[ 63 \div 3 = 21 \][/tex]
[tex]\[ 21 \div 3 = 7 \][/tex]

- The remaining number, 7, is already a prime.

We have now:
[tex]\[ 1008 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \][/tex]

2. Group the Prime Factors:
Next, we group the prime factors in pairs (as we are looking for a square root):

[tex]\[ (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times 7 \][/tex]

This can be rewritten using exponents:
[tex]\[ 1008 = 2^4 \times 3^2 \times 7 \][/tex]

3. Halve the Exponent:
To find the square root, we take half of each of the exponents in the factorization (where possible):

[tex]\[ \sqrt{1008} = \sqrt{2^4 \times 3^2 \times 7} = 2^{4/2} \times 3^{2/2} \times \sqrt{7} \][/tex]

Simplifying the exponents:
[tex]\[ \sqrt{1008} = 2^2 \times 3^1 \times \sqrt{7} = 4 \times 3 \times \sqrt{7} \][/tex]

Combining these:
[tex]\[ \sqrt{1008} = 12 \times \sqrt{7} \][/tex]

4. Estimate the Numerical Value:
Finally, we approximate the value of [tex]\(\sqrt{7}\)[/tex], which is roughly 2.64575. So:

[tex]\[ 12 \times \sqrt{7} \approx 12 \times 2.64575 = 31.749 \][/tex]

So, the square root of 1008 is approximately [tex]\(31.749\)[/tex]. The detailed steps confirm that the prime factors of 1008 are [tex]\(2, 2, 2, 2, 3, 3, 7\)[/tex]. The half counts product is 12, the remaining part inside the square root is 7, and incorporating this value, we get a final square root of approximately [tex]\(31.749\)[/tex].