To find the square root of 17,576 using the prime factorization method, we follow these steps:

Step 1: Prime Factorization of 17,576

First, divide 17,576 by the smallest prime number (2 in this case) until it can't be divided by 2 anymore.

$17,576 \div 2 = 8,788$ $8,788 \div 2 = 4,394$ $4,394 \div 2 = 2,197$ Since 2,197 is odd, we try dividing by the next smallest prime number, 3. It's not divisible by 3, so we try 7, 11, 13, 17, 19, and eventually:

$2,197 \div 13 = 169$ Now, 169 can be factored further:

$169 = 13 \times 13$

So, the prime factorization of 17,576 is:

$17,576 = 2^3 \times 13^2$

Step 2: Taking the Square Root

To find the square root, we pair the prime factors:

$\sqrt{17,576} = \sqrt{2^3 \times 13^2}$

For square root, we take one factor from each pair. From $2^3$, we can take one factor of 2 (since one pair is available), and from $13^2$, we take 13.

$\sqrt{17,576} = 2 \times 13 = 26$

Thus, the square root of 17,576 is 26.

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Related Questions:

1. How would you find the square root of 225 using the prime factorization method?
2. What is the prime factorization of 64, and how would you find its square root?
3. How does the prime factorization method compare to the long division method for square roots?
4. Can all numbers be factored into primes to find their square roots?
5. What is the cube root of 8,000 using the prime factorization method?

Tip:

Prime factorization is useful for finding exact roots, but for large numbers, approximation methods (like estimation) are often more practical.