Find the square roots of the numbers by expressing each number as the product of primes :
484

Answer

To find the square root of the number 484 by expressing it as the product of primes, we can follow these steps:

### Step-by-Step Solution:

1. **Prime Factorization of 484**:
- Start by dividing 484 by the smallest prime number, which is 2.
- \( 484 \div 2 = 242 \)
- \( 242 \div 2 = 121 \)
- Now, 121 is not divisible by 2, so we move to the next prime number, which is 3.
- 121 is not divisible by 3, so we try the next prime number, which is 5.
- 121 is also not divisible by 5, so we try 7.
- 121 is not divisible by 7, so we try 11.
- \( 121 \div 11 = 11 \)
- \( 11 \div 11 = 1 \)

Therefore, the prime factorization of 484 is:
\[ 484 = 2 \times 2 \times 11 \times 11 \]

2. **Expressing the Prime Factors**:
- We can express the prime factorization as:
\[ 484 = 2^2 \times 11^2 \]

3. **Finding the Square Root**:
- To find the square root of 484, we take the square root of each prime factor:
\[ \sqrt{484} = \sqrt{2^2 \times 11^2} \]
- Using the property of square roots, we can separate the factors:
\[ \sqrt{484} = \sqrt{2^2} \times \sqrt{11^2} \]
- This simplifies to:
\[ \sqrt{484} = 2 \times 11 \]

4. **Calculating the Final Answer**:
- Now, we multiply the results:
\[ 2 \times 11 = 22 \]

Thus, the square root of 484 is **22**.