To determine which choice is equivalent to the product [tex]\(\sqrt{x+2} \cdot \sqrt{x-2}\)[/tex], let's simplify the expression step by step:
1. Use properties of square roots: Recall that the product of two square roots can be written as the square root of the product of the two expressions inside the square roots:
[tex]\[
\sqrt{x+2} \cdot \sqrt{x-2} = \sqrt{(x+2)(x-2)}
\][/tex]
2. Simplify the product inside the square root: Multiply the two binomials inside the square root:
[tex]\[
(x+2)(x-2) = x^2 - 2^2 = x^2 - 4
\][/tex]
3. Substitute back into the square root:
[tex]\[
\sqrt{(x+2)(x-2)} = \sqrt{x^2 - 4}
\][/tex]
Therefore, the expression [tex]\(\sqrt{x+2} \cdot \sqrt{x-2}\)[/tex] simplifies to [tex]\(\sqrt{x^2 - 4}\)[/tex].
Now, let's compare this result with the given choices:
- A. [tex]\( x \)[/tex]
- B. [tex]\( \sqrt{x^2 + 4} \)[/tex]
- C. [tex]\( \sqrt{x^2} \)[/tex]
- D. [tex]\( \sqrt{x^2 - 4} \)[/tex]
The simplified expression [tex]\(\sqrt{x^2 - 4}\)[/tex] matches choice D.
Thus, the correct answer is:
D. [tex]\( \sqrt{x^2 - 4} \)[/tex]
