af 1764 . Write 1764 as a product of its prime factors. Use the prime factors to calculate \( \sqrt{1764} \)

To solve the problem, we'll follow two main steps: 1. **Factorize 1764 into its prime factors.** 2. **Use the prime factors to calculate \( \sqrt{1764} \).** --- ### 1. Prime Factorization of 1764 **Step-by-Step Breakdown:** 1. **Divide by 2 (the smallest prime number):** \[ 1764 \div 2 = 882 \] 2. **Divide by 2 again:** \[ 882 \div 2 = 441 \] 3. **441 is not even, so we try the next prime number, which is 3:** \[ 441 \div 3 = 147 \] 4. **Divide by 3 again:** \[ 147 \div 3 = 49 \] 5. **49 is not divisible by 3. Move to the next prime number, which is 7:** \[ 49 \div 7 = 7 \] 6. **Finally, divide by 7 once more:** \[ 7 \div 7 = 1 \] **Putting it all together:** \[ 1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7 \] This can be expressed using exponents as: \[ 1764 = 2^2 \times 3^2 \times 7^2 \] --- ### 2. Calculating \( \sqrt{1764} \) Using Prime Factors To find the square root of 1764 using its prime factors: \[ \sqrt{1764} = \sqrt{2^2 \times 3^2 \times 7^2} \] Since the square root of a product is the product of the square roots, and the square root of \( p^2 \) is \( p \), we have: \[ \sqrt{2^2} = 2 \\ \sqrt{3^2} = 3 \\ \sqrt{7^2} = 7 \] Multiply these together: \[ \sqrt{1764} = 2 \times 3 \times 7 = 42 \] --- **Answer:** The prime factorization of 1764 is \(2^2 \times 3^2 \times 7^2\). Using these factors, \( \sqrt{1764} = 42 \).