Find the square root of 8649 using the Prime factorization method?

To find the square root of 8649 using the prime factorization method, follow these steps:

Step 1: Find the prime factorization of 8649
Start by dividing 8649 by the smallest prime number, which is 2. Since 8649 is odd, it is not divisible by 2.

Next, divide by 3: 8649 ÷ 3 = 2883

Now, divide 2883 by 3 again: 2883 ÷ 3 = 961

961 is not divisible by 3, so move to the next prime number, which is 7. 961 is not divisible by 7, so try the next prime, 11.

961 ÷ 11 = 87.36, which is not an integer.

Now, try dividing 961 by 31 (since 961 is a known square): 961 ÷ 31 = 31

Thus, 961 = 31 × 31.

Step 2: Express the prime factorization
Now that we have the factors: 8649 = 3 × 3 × 31 × 31 8649 = 3² × 31²

Step 3: Take the square root
The square root of a number is the product of the square roots of its prime factors. From the prime factorization, we have: √8649 = √(3² × 31²)

Using the property of square roots, we get: √8649 = 3 × 31

Now, calculate the result: 3 × 31 = 93

Final Answer:
The square root of 8649 is 93.