What is the square root of 1764 using prime factorization?

The square root of 1764, when determined using the prime factorization method, is 42. This method efficiently finds the square root of a perfect square by breaking the number down into its most fundamental components.

Understanding Prime Factorization for Square Roots

Prime factorization is the process of breaking down a composite number into a product of its prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11).

When finding the square root of a number using this method, the key principle is that if a number is a perfect square, all its prime factors will appear an even number of times. To find the square root, you simply pair up identical prime factors and then take one factor from each pair.

For more information on prime factorization, you can refer to resources on number theory and prime numbers.

Step-by-Step Calculation for √1764

Let's break down how to find the square root of 1764 using prime factorization.

Step 1: Find the Prime Factorization of 1764

We systematically divide 1764 by the smallest possible prime numbers until the quotient becomes 1.

• Start with the smallest prime, 2:
  • 1764 ÷ 2 = 882
  • 882 ÷ 2 = 441
• Since 441 is not divisible by 2 (it's an odd number), try the next prime, 3:
  • 441 ÷ 3 = 147
  • 147 ÷ 3 = 49
• Since 49 is not divisible by 3 (4+9=13, not a multiple of 3), try the next prime, 5 (doesn't end in 0 or 5), then 7:
  • 49 ÷ 7 = 7
  • 7 ÷ 7 = 1

Thus, the prime factorization of 1764 is 2 × 2 × 3 × 3 × 7 × 7.

Step 2: Group the Prime Factors in Pairs

Once all prime factors are identified, group identical factors into pairs.

1764 = (2 × 2) × (3 × 3) × (7 × 7)

Step 3: Extract One Factor from Each Pair

For a square root, we take one number from each pair of identical prime factors.

• From (2 × 2), take 2
• From (3 × 3), take 3
• From (7 × 7), take 7

Step 4: Multiply the Extracted Factors

Finally, multiply the single factors extracted from each pair.

Square root of 1764 = 2 × 3 × 7 = 42

Prime Factorization of 1764: A Summary Table

This table summarizes the prime factorization process and the resulting pairs:

Therefore, the exact square root of 1764 using prime factorization is 42. This method is particularly useful for understanding the composition of numbers and confirming that a number is a perfect square.