To solve the problem of expressing the product [tex]\(\frac{3 w x}{6 x} \cdot \frac{3 w x}{9 w}\)[/tex] in its simplest form, let's go through the step-by-step simplification process.
### Step 1: Simplify Each Fraction
1. Simplifying [tex]\(\frac{3 w x}{6 x}\)[/tex]:
- The numerator is [tex]\(3 w x\)[/tex].
- The denominator is [tex]\(6 x\)[/tex].
Since [tex]\(x\)[/tex] is present in both the numerator and the denominator, we can cancel [tex]\(x\)[/tex] from both, giving us:
[tex]\[
\frac{3 w x}{6 x} = \frac{3 w}{6} = \frac{1}{2} w
\][/tex]
2. Simplifying [tex]\(\frac{3 w x}{9 w}\)[/tex]:
- The numerator is [tex]\(3 w x\)[/tex].
- The denominator is [tex]\(9 w\)[/tex].
Since [tex]\(w\)[/tex] is present in both the numerator and the denominator, we can cancel [tex]\(w\)[/tex] from both, giving us:
[tex]\[
\frac{3 w x}{9 w} = \frac{3 x}{9} = \frac{1}{3} x
\][/tex]
### Step 2: Multiply the Simplified Fractions
Now, we multiply the simplified fractions:
[tex]\[
\left(\frac{1}{2} w\right) \cdot \left(\frac{1}{3} x\right)
\][/tex]
Multiply the numerators together and the denominators together:
[tex]\[
\frac{1 \cdot 1 \cdot w \cdot x}{2 \cdot 3} = \frac{w x}{6}
\][/tex]
Hence, the product [tex]\(\frac{3 w x}{6 x} \cdot \frac{3 w x}{9 w}\)[/tex] simplifies to [tex]\(\frac{w x}{6}\)[/tex].
### Conclusion
The expression [tex]\(\frac{3 w x}{6 x} \cdot \frac{3 w x}{9 w}\)[/tex] in its simplest form is:
[tex]\[
\frac{w x}{6}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{w x}{6}}
\][/tex]
