To simplify the given expression, let's break it down step-by-step:
The expression is:
[tex]\[
\left( \frac{2n}{6n+4} \right) \left( \frac{3n+2}{3n-2} \right)
\][/tex]
1. Factor the denominators and numerators where possible:
- In the fraction [tex]\(\frac{2n}{6n + 4}\)[/tex], notice that [tex]\(6n + 4\)[/tex] can be factored:
[tex]\[
6n + 4 = 2(3n + 2)
\][/tex]
So, the fraction becomes:
[tex]\[
\frac{2n}{2(3n + 2)}
\][/tex]
2. Simplify the expression:
- We can cancel the common factor of 2 in the numerator and the denominator:
[tex]\[
\frac{2n}{2(3n + 2)} = \frac{n}{3n + 2}
\][/tex]
3. Now, let's multiply the simplified fraction with the second fraction:
[tex]\[
\left( \frac{n}{3n + 2} \right) \left( \frac{3n + 2}{3n - 2} \right)
\][/tex]
4. Simplify the product:
- Notice that [tex]\((3n + 2)\)[/tex] in the numerator and the denominator will cancel out:
[tex]\[
\frac{n}{3n + 2} \cdot \frac{3n + 2}{3n - 2} = \frac{n \cdot (3n + 2)}{(3n + 2) \cdot (3n - 2)}
\][/tex]
- This can be simplified further as:
[tex]\[
\frac{n}{3n - 2}
\][/tex]
However, based on the calculations provided, it is clear that the numerator of the simplified expression directly turns out to be:
[tex]\[
n
\][/tex]
Thus, the numerator of the simplified expression is:
[tex]\[
\boxed{n}
\][/tex]
