Find the square root of 441 by prime factorization method. Explain how the answer is obtained.
The correct answer and explanation is:
To find the square root of 441 using the prime factorization method, the first step is to break down 441 into its prime factors.
Step 1: Begin the Prime Factorization
Start by dividing 441 by the smallest prime number, which is 2. However, since 441 is an odd number, it cannot be divided by 2. The next smallest prime number is 3.
441 ÷ 3 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1
So, the prime factorization of 441 is:
441 = 3 × 3 × 7 × 7
Step 2: Pair the Prime Factors
The prime factorization of 441 shows that 441 = 3² × 7². To find the square root, we take one factor from each pair of identical prime factors.
- From the pair of 3’s (3²), take one 3.
- From the pair of 7’s (7²), take one 7.
Step 3: Multiply the Resulting Factors
Now, multiply the numbers obtained in step 2:
√441 = 3 × 7 = 21
Final Answer:
The square root of 441 is 21.
Explanation:
The prime factorization method works by breaking down the number into its prime factors, then pairing up identical factors. Once the pairs are identified, you can take one factor from each pair and multiply them together to obtain the square root. This method is very useful because it ensures the number is completely reduced to prime factors, making it easier to find square roots. In the case of 441, the prime factorization reveals two pairs of identical factors (3² and 7²), so the square root of 441 is simply the product of one 3 and one 7, which equals 21.