To simplify the given expression:
[tex]\[
\left( \frac{2n}{6n + 4} \right) \left( \frac{3n + 2}{3n - 2} \right)
\][/tex]
we can follow these steps:
### Step 1: Simplify each fraction separately
#### Simplify the first fraction [tex]\(\frac{2n}{6n + 4}\)[/tex]:
1. Factor out the greatest common divisor (GCD) in the denominator:
[tex]\[
6n + 4 = 2(3n + 2)
\][/tex]
2. Rewrite the fraction:
[tex]\[
\frac{2n}{2(3n + 2)} = \frac{2n}{2} \cdot \frac{1}{3n + 2}
\][/tex]
3. Simplify by canceling the common factor of 2:
[tex]\[
\frac{n}{3n + 2}
\][/tex]
So, the simplified form of the first fraction is:
[tex]\[
\frac{n}{3n + 2}
\][/tex]
#### Simplify the second fraction [tex]\(\frac{3n + 2}{3n - 2}\)[/tex]:
1. The second fraction is already in its simplest form, so we leave it as:
[tex]\[
\frac{3n + 2}{3n - 2}
\][/tex]
### Step 2: Multiply the simplified fractions
Now, multiply the simplified fractions:
[tex]\[
\left( \frac{n}{3n + 2} \right) \left( \frac{3n + 2}{3n - 2} \right)
\][/tex]
1. The [tex]\(3n + 2\)[/tex] terms in the numerator of the second fraction and the denominator of the first fraction will cancel each other out:
[tex]\[
\frac{n \cdot (3n + 2)}{(3n + 2) \cdot (3n - 2)} = \frac{n}{3n - 2}
\][/tex]
### Step 3: Identify the numerator of the simplified expression
The final simplified fraction is:
[tex]\[
\frac{n}{3n - 2}
\][/tex]
Thus, the numerator of the simplified expression is:
[tex]\[
n
\][/tex]
