To express [tex]\( \sqrt{45} \)[/tex] in its simplest radical form, we need to follow a series of steps involving factorization and simplification of the square root.
### Step-by-Step Solution:
1. Factorize the Number Inside the Square Root:
First, we need to find the factors of 45. Notice that:
[tex]\[
45 = 9 \times 5
\][/tex]
Therefore, we can write:
[tex]\[
\sqrt{45} = \sqrt{9 \times 5}
\][/tex]
2. Use the Property of Square Roots:
We use the property of square roots that the square root of a product is the product of the square roots:
[tex]\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}
\][/tex]
3. Simplify the Square Roots:
We know that [tex]\( \sqrt{9} \)[/tex] is a perfect square, specifically:
[tex]\[
\sqrt{9} = 3
\][/tex]
Therefore, we can simplify:
[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]
