To determine which expression is equivalent to [tex]\(\sqrt{45}\)[/tex], let's simplify the radical expression step by step.
1. Factor the number inside the square root:
We start with [tex]\( \sqrt{45} \)[/tex]. To simplify this, we look for factors of 45 that are perfect squares.
[tex]\[
45 = 9 \times 5
\][/tex]
2. Use the property of square roots:
The property [tex]\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)[/tex] allows us to break the original square root into two separate square roots.
[tex]\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}
\][/tex]
3. Simplify the square root of the perfect square:
Since [tex]\( \sqrt{9} = 3 \)[/tex], we can further simplify:
[tex]\[
\sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5}
\][/tex]
4. Combine the simplified terms:
Therefore, the expression [tex]\( \sqrt{45} \)[/tex] simplifies to:
[tex]\[
\sqrt{45} = 3 \sqrt{5}
\][/tex]
5. Identify the correct answer:
Comparing this with the given multiple-choice options:
- A. [tex]\( 9 \sqrt{5} \)[/tex] is not equivalent.
- B. [tex]\( 5 \sqrt{3} \)[/tex] is not equivalent.
- C. [tex]\( 3 \sqrt{5} \)[/tex] is equivalent.
- D. [tex]\( 5 \sqrt{9} \)[/tex] is not equivalent.
Thus, the correct answer is:
[tex]\[
\boxed{3 \sqrt{5}}
\][/tex]
