To determine which of the given choices is equivalent to the product [tex]\(\sqrt{x+3} \cdot \sqrt{x-3}\)[/tex], let's go through the simplification process step-by-step.
1. Understand the product of square roots:
[tex]\[
\sqrt{x+3} \cdot \sqrt{x-3}
\][/tex]
2. Use the property of square roots:
The property of square roots that is useful here is:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
Applying this property:
[tex]\[
\sqrt{x+3} \cdot \sqrt{x-3} = \sqrt{(x+3)(x-3)}
\][/tex]
3. Simplify the expression inside the square root:
We need to simplify the expression [tex]\((x+3)(x-3)\)[/tex]. This is a difference of squares:
[tex]\[
(x+3)(x-3) = x^2 - 9
\][/tex]
4. Write the simplified form:
Now, substituting back, we have:
[tex]\[
\sqrt{(x+3)(x-3)} = \sqrt{x^2 - 9}
\][/tex]
So, the product [tex]\(\sqrt{x+3} \cdot \sqrt{x-3}\)[/tex] simplifies to [tex]\(\sqrt{x^2 - 9}\)[/tex].
5. Determine the correct choice:
Among the choices given:
A. [tex]\(\sqrt{x^2}\)[/tex]
B. [tex]\(\sqrt{x^2 + 9}\)[/tex]
C. [tex]\(x\)[/tex]
D. [tex]\(\sqrt{x^2 - 9}\)[/tex]
The expression [tex]\(\sqrt{x^2 - 9}\)[/tex] matches choice D.
Therefore, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
