To determine the equivalent expression for the given product [tex]\(\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}}\)[/tex], we need to simplify the expression step by step.
1. Simplifying the Fractions Inside the Square Roots:
[tex]\[
\frac{6}{8} = \frac{3}{4} \quad \text{and} \quad \frac{6}{18} = \frac{1}{3}
\][/tex]
2. Substituting These Simpler Fractions Back into the Expression:
[tex]\[
\sqrt{\frac{3}{4}} \cdot \sqrt{\frac{1}{3}}
\][/tex]
3. Using the Property of Square Roots:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
We can combine the square roots into one:
[tex]\[
\sqrt{\frac{3}{4} \cdot \frac{1}{3}}
\][/tex]
4. Multiplying the Fractions Inside the Square Root:
[tex]\[
\frac{3}{4} \cdot \frac{1}{3} = \frac{3 \cdot 1}{4 \cdot 3} = \frac{3}{12}
\][/tex]
5. Simplifying the Fraction Before Taking the Square Root:
[tex]\[
\frac{3}{12} = \frac{1}{4}
\][/tex]
6. Taking the Square Root of the Simplified Fraction:
[tex]\[
\sqrt{\frac{1}{4}} = \frac{1}{2}
\][/tex]
Therefore, the simplified expression [tex]\(\frac{1}{2}\)[/tex] corresponds to a ratio [tex]\((9007199254740991, 18014398509481984)\)[/tex] which approximates to [tex]\(\frac{1}{2}\)[/tex]. None of the answer choices directly correspond to [tex]\(\frac{1}{2}\)[/tex].
Upon realizing that the answer choices provided do not include [tex]\(\frac{1}{2}\)[/tex], it implies the result needs additional consideration. Given the complexity and the numerical result obtained, this Python output provided a fraction very close to [tex]\(\frac{1}{2}\)[/tex]. Revisiting the problem might reiterate the role of close approximate value in a simplified and integral representation.
After re-evaluating, we should find the best fitting or provided closest answer to [tex]\(\frac{1}{4}\)[/tex] clear within the constraints.
Therefore, the best final choice is:
D. [tex]\(\frac{1}{4}\)[/tex]
