Certainly! Let's solve the problem step-by-step.
### Prime Factorization of 576
To begin with, we need to find the prime factorization of 576. We repeatedly divide by the smallest prime number, which is 2, until we no longer can:
1. [tex]\( 576 \div 2 = 288 \)[/tex]
2. [tex]\( 288 \div 2 = 144 \)[/tex]
3. [tex]\( 144 \div 2 = 72 \)[/tex]
4. [tex]\( 72 \div 2 = 36 \)[/tex]
5. [tex]\( 36 \div 2 = 18 \)[/tex]
6. [tex]\( 18 \div 2 = 9 \)[/tex]
Next, we divide by the next smallest prime, which is 3:
7. [tex]\( 9 \div 3 = 3 \)[/tex]
8. [tex]\( 3 \div 3 = 1 \)[/tex]
So, the prime factorization of 576 is:
[tex]\[ 576 = 2^6 \times 3^2 \][/tex]
### Prime Factorization of 64
Similarly, we find the prime factorization of 64:
1. [tex]\( 64 \div 2 = 32 \)[/tex]
2. [tex]\( 32 \div 2 = 16 \)[/tex]
3. [tex]\( 16 \div 2 = 8 \)[/tex]
4. [tex]\( 8 \div 2 = 4 \)[/tex]
5. [tex]\( 4 \div 2 = 2 \)[/tex]
6. [tex]\( 2 \div 2 = 1 \)[/tex]
So, the prime factorization of 64 is:
[tex]\[ 64 = 2^6 \][/tex]
### Simplifying the Expression [tex]\(\sqrt{\frac{576}{64}}\)[/tex]
We now simplify the expression:
[tex]\[ \sqrt{\frac{576}{64}} \][/tex]
First, note that:
[tex]\[ \frac{576}{64} = \frac{2^6 \times 3^2}{2^6} \][/tex]
We can cancel out [tex]\(2^6\)[/tex] in the numerator and denominator:
[tex]\[ \frac{2^6 \times 3^2}{2^6} = 3^2 = 9 \][/tex]
Thus:
[tex]\[ \sqrt{\frac{576}{64}} = \sqrt{9} \][/tex]
Finally, we take the square root of 9:
[tex]\[ \sqrt{9} = 3 \][/tex]
### Summary of Solutions
- The prime factorization of 576 is [tex]\(2^6 \times 3^2\)[/tex].
- The prime factorization of 64 is [tex]\(2^6\)[/tex].
- The expression [tex]\(\sqrt{\frac{576}{64}}\)[/tex] in simplest form is [tex]\(3\)[/tex].
Therefore, the simplified form of [tex]\(\sqrt{\frac{576}{64}}\)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
