To express \(\sqrt{150}\) in simplest radical form, we follow these steps:
1. Factor the number under the square root:
We start by factoring 150 into its prime factors.
[tex]\[
150 = 2 \times 3 \times 5^2
\][/tex]
2. Group factors into pairs:
We look for pairs of equal factors. In this case, \(5^2\) is a pair of equal factors.
3. Simplify using pairs:
According to the properties of square roots, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Apply this property to the factor pairs:
[tex]\[
\sqrt{150} = \sqrt{5^2 \times 6} = \sqrt{5^2} \times \sqrt{6} = 5 \times \sqrt{6}
\][/tex]
So, the simplest radical form of \(\sqrt{150}\) is:
[tex]\[
5\sqrt{6}
\][/tex]
If we were to evaluate [tex]\(5\sqrt{6}\)[/tex], it would give us approximately 12.24744871391589.
