Which choice is equivalent to the product below for acceptable values of [tex]$x$[/tex]?

[tex] \sqrt{x+3} \cdot \sqrt{x-3} [/tex]

A. [tex] \sqrt{x^2-9} [/tex]
B. [tex] \sqrt{x^2} [/tex]
C. [tex] x [/tex]
D. [tex] \sqrt{x^2+9} [/tex]



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.