First, let's carefully analyze the given expression [tex]\(\sqrt{x+3} \cdot \sqrt{x-3}\)[/tex].
The property of square roots that we use here is that the product of the square roots is the same as the square root of the product, i.e., [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex].
Applying this property to our expression:
[tex]\[
\sqrt{x+3} \cdot \sqrt{x-3} = \sqrt{(x+3) \cdot (x-3)}
\][/tex]
Next, we need to simplify the expression inside the square root:
[tex]\[
(x+3) \cdot (x-3)
\][/tex]
This is a difference of squares, which can be factored by the formula [tex]\(a^2 - b^2 = (a+b)(a-b)\)[/tex]. Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex], so:
[tex]\[
(x+3)(x-3) = x^2 - 3^2 = x^2 - 9
\][/tex]
Thus, our expression simplifies to:
[tex]\[
\sqrt{(x+3) \cdot (x-3)} = \sqrt{x^2 - 9}
\][/tex]
Therefore, the choice that is equivalent to the product [tex]\(\sqrt{x+3} \cdot \sqrt{x-3}\)[/tex] for acceptable values of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{\sqrt{x^2 - 9}}
\][/tex]
Hence, the correct choice is A.
