Let's simplify the expression step-by-step to determine which of the given choices is equivalent to the product when [tex]\( x > 0 \)[/tex]:
Given:
[tex]\[ \sqrt{\frac{2}{x}} \cdot \sqrt{\frac{x^2}{8}} \][/tex]
First, we use the property of square roots that states [tex]\( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)[/tex]. Therefore, we can combine the square roots:
[tex]\[ \sqrt{\frac{2}{x}} \cdot \sqrt{\frac{x^2}{8}} = \sqrt{\left(\frac{2}{x}\right) \cdot \left(\frac{x^2}{8}\right)} \][/tex]
Next, we multiply the expressions inside the square root:
[tex]\[ \left(\frac{2}{x}\right) \cdot \left(\frac{x^2}{8}\right) = \frac{2 \cdot x^2}{x \cdot 8} = \frac{2x^2}{8x} = \frac{x}{4} \][/tex]
So, now we have:
[tex]\[ \sqrt{\frac{2}{x}} \cdot \sqrt{\frac{x^2}{8}} = \sqrt{\frac{x}{4}} \][/tex]
Now, we look at the given answer choices and see which one matches [tex]\( \sqrt{\frac{x}{4}} \)[/tex].
A. [tex]\( \frac{x}{4} \)[/tex]
B. [tex]\( \frac{\sqrt{x}}{2} \)[/tex]
C. [tex]\( \frac{x}{2} \)[/tex]
D. [tex]\( \sqrt{\frac{x}{2}} \)[/tex]
It is clear that none of the answer choices matches [tex]\( \sqrt{\frac{x}{4}} \)[/tex].
Therefore, the correct answer is:
[tex]\[ \text{None of the given choices are equivalent to the product} \][/tex]
So, the answer is:
None
