To solve the problem of finding an equivalent expression to [tex]\(\sqrt{45}\)[/tex], let's go step-by-step through the process of simplifying the square root.
1. Recognize that 45 is not a perfect square:
[tex]\[
\sqrt{45}
\][/tex]
2. Find the prime factorization of 45:
[tex]\[
45 = 3 \times 3 \times 5
\][/tex]
3. Use properties of square roots to simplify:
[tex]\[
\sqrt{45} = \sqrt{3 \times 3 \times 5}
\][/tex]
Since [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we can rewrite this as:
[tex]\[
\sqrt{3 \times 3 \times 5} = \sqrt{3^2 \times 5}
\][/tex]
4. Take the square root of the perfect square (3^2):
[tex]\[
\sqrt{3^2 \times 5} = \sqrt{3^2} \times \sqrt{5} = 3 \times \sqrt{5}
\][/tex]
Thus, [tex]\(\sqrt{45}\)[/tex] simplifies to [tex]\(3 \sqrt{5}\)[/tex].
Among the given options:
- A. [tex]\(5 \sqrt{9}\)[/tex] is equivalent to [tex]\(5 \times 3 = 15\)[/tex]
- a. [tex]\(5 \sqrt{3}\)[/tex] is incorrect because it does not match our simplified form
- C. [tex]\(9 \sqrt{5}\)[/tex] is incorrect because it suggests a different multiplication factor
- D. [tex]\(3 \sqrt{5}\)[/tex] matches exactly with our simplified expression
The correct answer is:
[tex]\[
\boxed{D. \, 3 \sqrt{5}}
\][/tex]
