To simplify the expression [tex]\(\sqrt{\frac{576}{64}}\)[/tex], we need to follow a series of steps:
1. Prime Factorization of 576:
- The number 576 can be broken down into its prime factors. After systematic factorization, we find:
- [tex]\(576 = 2^6 \times 3^2\)[/tex]
- Therefore, the prime factorization of 576 is [tex]\([2, 2, 2, 2, 2, 2, 3, 3]\)[/tex].
2. Prime Factorization of 64:
- Similarly, breaking down 64 into prime factors gives us:
- [tex]\(64 = 2^6\)[/tex]
- Therefore, the prime factorization of 64 is [tex]\([2, 2, 2, 2, 2, 2]\)[/tex].
3. Simplifying the Expression:
- Now, we simplify the expression [tex]\(\sqrt{\frac{576}{64}}\)[/tex].
- Notice that [tex]\(\frac{576}{64} = \frac{2^6 \times 3^2}{2^6}\)[/tex].
- When we simplify this, the [tex]\(2^6\)[/tex] terms cancel out:
- [tex]\(\frac{576}{64} = 3^2 = 9\)[/tex].
4. Taking the Square Root:
- Finally, we take the square root of the simplified fraction:
- [tex]\(\sqrt{9} = 3\)[/tex].
Thus, the prime factorization of 576 is [tex]\([2, 2, 2, 2, 2, 2, 3, 3]\)[/tex], the prime factorization of 64 is [tex]\([2, 2, 2, 2, 2, 2]\)[/tex], and the expression [tex]\(\sqrt{\frac{576}{64}}\)[/tex] in simplest form is [tex]\(3.0\)[/tex].
