To express [tex]\(\sqrt{150}\)[/tex] in simplest radical form, we need to identify the prime factorization of the number 150 and simplify accordingly.
1. Prime Factorization of 150:
- The number 150 can be factored into prime numbers:
[tex]\[
150 = 2 \times 3 \times 5^2
\][/tex]
2. Group Factors into Pairs:
- We can pair the factors to identify the squares that can be taken out from under the radical. Notice that we have:
[tex]\[
5^2 \text{ which is a perfect square.}
\][/tex]
- So, we can separate [tex]\(\sqrt{150}\)[/tex] into:
[tex]\[
\sqrt{2 \times 3 \times 5^2}
\][/tex]
3. Simplify the Expression:
- We can take 5 out of the radical because it is a perfect square:
[tex]\[
\sqrt{2 \times 3 \times 5^2} = \sqrt{2 \times 3} \times \sqrt{5^2} = \sqrt{6} \times 5 = 5 \sqrt{6}
\][/tex]
Therefore, the simplest radical form of [tex]\(\sqrt{150}\)[/tex] is:
[tex]\[
\boxed{5 \sqrt{6}}
\][/tex]
Additionally, if you approximate the value, [tex]\(5\sqrt{6}\)[/tex] is approximately:
[tex]\[
5 \times 2.44949 = 12.24744871391589
\][/tex]
The detailed answer is:
[tex]\[
\boxed{5 \sqrt{6} \approx 12.24744871391589}
\][/tex]
