Let's break down the given expression and find its equivalent value step by step:
1. Calculate the product of the square roots:
[tex]\[
\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}}
\][/tex]
2. We know that the product of two square roots can be combined under a single square root:
[tex]\[
\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}} = \sqrt{\left(\frac{6}{8}\right) \cdot \left(\frac{6}{18}\right)}
\][/tex]
3. Multiply the fractions inside the square root:
[tex]\[
\left(\frac{6}{8}\right) \cdot \left(\frac{6}{18}\right) = \frac{6 \cdot 6}{8 \cdot 18} = \frac{36}{144}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{36}{144} = \frac{1}{4}
\][/tex]
5. Now, take the square root of the simplified fraction:
[tex]\[
\sqrt{\frac{1}{4}} = \frac{1}{2}
\][/tex]
So, the product [tex]\(\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}} = \frac{1}{2}\)[/tex].
Now let's compare this result with the given choices:
- A. [tex]\(\frac{7}{12}\)[/tex]
- B. [tex]\(\frac{3}{4}\)[/tex]
- C. [tex]\(\frac{6}{12}\)[/tex] which simplifies to [tex]\(\frac{1}{2}\)[/tex]
- D. [tex]\(\frac{1}{4}\)[/tex]
- E. [tex]\(\frac{7}{1}\)[/tex]
From the above choices, it's clear that:
[tex]\[
\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}} = \frac{1}{2}
\][/tex]
Thus, the equivalent choice is:
[tex]\[
\boxed{\frac{6}{12}}
\][/tex]
which simplifies to [tex]\(\boxed{C}\)[/tex].
