To simplify the square root of 147, we can follow these steps:
1. Prime Factorization:
- Start by performing the prime factorization of 147. Break down 147 into its prime factors.
- 147 is divisible by 3 (since [tex]\( 1 + 4 + 7 = 12 \)[/tex]), so:
[tex]\[
147 \div 3 = 49
\][/tex]
- Now factor 49. Since 49 is [tex]\( 7 \times 7 \)[/tex], we have:
[tex]\[
49 = 7^2
\][/tex]
Therefore, the prime factorization of 147 is:
[tex]\[
147 = 3 \times 7^2
\][/tex]
2. Simplify the Square Root:
- Now apply the square root to the prime factorization:
[tex]\[
\sqrt{147} = \sqrt{3 \times 7^2}
\][/tex]
- Since the square root of a product is the product of the square roots, we can separate the terms:
[tex]\[
\sqrt{147} = \sqrt{3} \times \sqrt{7^2}
\][/tex]
- Evaluate the square root of [tex]\(7^2\)[/tex]:
[tex]\[
\sqrt{7^2} = 7
\][/tex]
- Combine the simplified terms:
[tex]\[
\sqrt{147} = 7 \times \sqrt{3}
\][/tex]
So, the simplest radical form of [tex]\(\sqrt{147}\)[/tex] is:
[tex]\[
7\sqrt{3}
\][/tex]
To provide you with the numerical result:
The approximate value of [tex]\(7\sqrt{3}\)[/tex] is 12.12435565298214 when calculated.
