By splitting into prime factors, find the square root of :
`396900`

Options

A. `640`
B. `630`
C. `360`
D. `603`

Answer

To find the square root of 396900 using the prime factorization method, we will follow these steps:

### Step-by-Step Solution:

1. **Start with the number**: We have the number 396900.

2. **Divide by the smallest prime number**: We begin by dividing the number by the smallest prime number, which is 2. Since 396900 is even, we can divide it by 2.
\[ 396900 \div 2 = 198450 \]

3. **Continue dividing by 2**: We can divide 198450 by 2 again since it is still even.
\[ 198450 \div 2 = 99225 \]

4. **Next prime number**: Now, we will try dividing by 3, as 99225 is not divisible by 2.
\[ 99225 \div 3 = 33075 \]

5. **Continue with 3**: We can divide 33075 by 3 again.
\[ 33075 \div 3 = 11025 \]

6. **Continue with 3**: We can divide 11025 by 3 again.
\[ 11025 \div 3 = 3675 \]

7. **Continue with 3**: We can divide 3675 by 3 again.
\[ 3675 \div 3 = 1225 \]

8. **Next prime number**: Now, we will try dividing by 5, as 1225 is not divisible by 3.
\[ 1225 \div 5 = 245 \]

9. **Continue with 5**: We can divide 245 by 5 again.
\[ 245 \div 5 = 49 \]

10. **Next prime number**: Now, we will divide 49 by 7, as it is not divisible by 5.
\[ 49 \div 7 = 7 \]

11. **Final division**: Finally, we divide 7 by 7.
\[ 7 \div 7 = 1 \]

12. **List the prime factors**: Now we have the complete prime factorization of 396900:
\[ 396900 = 2^2 \times 3^4 \times 5^2 \times 7^2 \]

13. **Finding the square root**: To find the square root, we take the square root of each prime factor:
\[ \sqrt{396900} = \sqrt{(2^2) \times (3^4) \times (5^2) \times (7^2)} = 2^{2/2} \times 3^{4/2} \times 5^{2/2} \times 7^{2/2} \]
\[ = 2^1 \times 3^2 \times 5^1 \times 7^1 = 2 \times 9 \times 5 \times 7 \]

14. **Calculating the final result**:
\[ = 2 \times 9 = 18 \]
\[ = 18 \times 5 = 90 \]
\[ = 90 \times 7 = 630 \]

Thus, the square root of 396900 is **630**.