To simplify [tex]\(\sqrt{45}\)[/tex] in the simplest radical form, let's follow these steps:
1. Factor the number under the square root sign into its prime factors:
- [tex]\(45 = 9 \times 5\)[/tex]
- Notice that [tex]\(9\)[/tex] is a perfect square because [tex]\(9 = 3^2\)[/tex].
2. Rewrite the square root expression using the factorization:
- [tex]\(\sqrt{45} = \sqrt{9 \times 5}\)[/tex]
3. Use the property of square roots that states [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
- [tex]\(\sqrt{45} = \sqrt{9} \times \sqrt{5}\)[/tex]
4. Simplify the square root of the perfect square:
- [tex]\(\sqrt{9} = 3\)[/tex]
5. Combine the simplified square root with the remaining factor:
- [tex]\(\sqrt{45} = 3 \times \sqrt{5}\)[/tex]
So, the simplest radical form of [tex]\(\sqrt{45}\)[/tex] is:
[tex]\[
\boxed{3\sqrt{5}}
\][/tex]
