To find the simplest form of [tex]\(\sqrt{192}\)[/tex], let's follow these steps carefully.
Step 1: Prime Factorization of 192
First, we perform the prime factorization of 192:
1. [tex]\(192 \div 2 = 96\)[/tex]
2. [tex]\(96 \div 2 = 48\)[/tex]
3. [tex]\(48 \div 2 = 24\)[/tex]
4. [tex]\(24 \div 2 = 12\)[/tex]
5. [tex]\(12 \div 2 = 6\)[/tex]
6. [tex]\(6 \div 2 = 3\)[/tex]
7. 3 is a prime number that cannot be divided by 2, so [tex]\(3\)[/tex] is left as it is.
Thus, the prime factorization of 192 is:
[tex]\[192 = 2^6 \cdot 3\][/tex]
Step 2: Simplify the Square Root
Now, we want to simplify [tex]\(\sqrt{192}\)[/tex] using the prime factors:
[tex]\[
\sqrt{192} = \sqrt{2^6 \cdot 3}
\][/tex]
To simplify [tex]\(\sqrt{2^6 \cdot 3}\)[/tex], we treat the square root of a product as the product of square roots:
[tex]\[
\sqrt{2^6 \cdot 3} = \sqrt{2^6} \cdot \sqrt{3}
\][/tex]
Next, we simplify [tex]\(\sqrt{2^6}\)[/tex]:
[tex]\[
\sqrt{2^6} = 2^{6/2} = 2^3 = 8
\][/tex]
Thus:
[tex]\[
\sqrt{192} = 8 \cdot \sqrt{3}
\][/tex]
Step 3: Choose the Correct Answer
We have simplified [tex]\(\sqrt{192}\)[/tex] to [tex]\(8 \sqrt{3}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(8 \sqrt{3}\)[/tex]
