Find the square root of 484 through decomposition into prime factors.
The correct answer and explanation is:
To find the square root of 484 through prime factorization, the first step is to decompose the number 484 into its prime factors.
Step 1: Divide by the smallest prime number
Start with dividing 484 by 2, the smallest prime number.
484 ÷ 2 = 242
Step 2: Divide again by 2
Now, divide 242 by 2.
242 ÷ 2 = 121
Step 3: Check for divisibility by the next prime number
121 is not divisible by 2, so we move to the next smallest prime number, which is 3. Since 121 is not divisible by 3, we try the next prime number, 5. 121 is not divisible by 5 either. Moving to the next prime number, 7, 11, and finally 11.
121 ÷ 11 = 11
Step 4: Prime Factorization of 484
Now we have successfully decomposed 484 into prime factors:
484 = 2 × 2 × 11 × 11
Step 5: Pair the prime factors
To take the square root, pair the prime factors. In this case, we have two pairs of 2s and 11s. So, the square root of 484 is found by taking one number from each pair.
√484 = √(2 × 2 × 11 × 11) = 2 × 11 = 22
Conclusion
The square root of 484 is 22. This method of prime factorization works because squaring the number 22 gives 484, verifying that the square root is indeed correct.