To determine which choice is equivalent to the product [tex]\(\sqrt{2} \cdot \sqrt{8} \cdot \sqrt{4}\)[/tex], let's simplify the expression step-by-step.
1. Simplify the square roots:
- [tex]\(\sqrt{2}\)[/tex] remains as [tex]\(\sqrt{2}\)[/tex].
- [tex]\(\sqrt{8}\)[/tex] can be simplified. Since [tex]\(8 = 4 \times 2\)[/tex], this becomes [tex]\(\sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\)[/tex].
- [tex]\(\sqrt{4}\)[/tex] simplifies directly to 2, because [tex]\(4 = 2^2\)[/tex] and the square root of 4 is 2.
2. Rewrite the product using these simplified forms:
[tex]\[
\sqrt{2} \cdot \sqrt{8} \cdot \sqrt{4} = \sqrt{2} \cdot (2\sqrt{2}) \cdot 2
\][/tex]
3. Combine the terms:
- First, combine [tex]\(\sqrt{2}\)[/tex] and [tex]\(2\sqrt{2}\)[/tex]:
[tex]\[
\sqrt{2} \cdot 2\sqrt{2} = 2 \cdot (\sqrt{2} \cdot \sqrt{2}) = 2 \cdot 2 = 4
\][/tex]
- Then, multiply this result by 2:
[tex]\[
4 \cdot 2 = 8
\][/tex]
Thus, the expression [tex]\(\sqrt{2} \cdot \sqrt{8} \cdot \sqrt{4}\)[/tex] simplifies to 8.
Therefore, the correct choice is [tex]\( \text{D. } 8 \)[/tex].
