54n/32n*2n-2

To simplify the expression \( \frac{54n}{32n \cdot 2n - 2} \), let's first simplify the denominator:

1. Simplify the denominator: \[ 32n \cdot 2n - 2 = 64n^2 - 2 \] Therefore, the expression now looks like this: \[ \frac{54n}{64n^2 - 2} \]

2. Factor the denominator: Notice that \( 64n^2 - 2 \) can be factored out: \[ = 2(32n^2 - 1) = 2(4n - 1)(4n + 1) \quad \text{(using the difference of squares)} \]

3. Rewrite the expression: Now, we can rewrite the full expression: \[ \frac{54n}{2(4n - 1)(4n + 1)} \]

4. Simplify the fraction: The \( 2 \) in the denominator can be simplified with \( 54 \): \[ = \frac{54n}{2(4n - 1)(4n + 1)} = \frac{27n}{(4n - 1)(4n + 1)} \]

So, the simplified expression is: \[ \frac{27n}{(4n - 1)(4n + 1)} \]