To simplify [tex]\(\sqrt{147}\)[/tex], let's follow a step-by-step approach.
1. Prime Factorization of 147:
- We can start by breaking down the number 147 using its prime factors.
- 147 can be divided by 3 (since the sum of its digits, 1+4+7 = 12, is divisible by 3):
[tex]\[
147 \div 3 = 49
\][/tex]
- Next, 49 is a perfect square:
[tex]\[
49 = 7 \times 7
\][/tex]
- So, the prime factorization of 147 is:
[tex]\[
147 = 3 \times 7 \times 7 = 3 \times 7^2
\][/tex]
2. Applying the Square Root:
- Using the property of square roots, we can rewrite [tex]\(\sqrt{147}\)[/tex] as:
[tex]\[
\sqrt{147} = \sqrt{3 \times 7^2}
\][/tex]
- We can separate the square root over the factors:
[tex]\[
\sqrt{3 \times 7^2} = \sqrt{3} \times \sqrt{7^2}
\][/tex]
- Since [tex]\(\sqrt{7^2} = 7\)[/tex], this simplifies to:
[tex]\[
\sqrt{3} \times 7 = 7 \sqrt{3}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{147}\)[/tex] is [tex]\(7 \sqrt{3}\)[/tex].
The correct answer is:
C. [tex]\(7 \sqrt{3}\)[/tex]
