Answer :
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.
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.