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: \[ \frac{27n}{(4n - 1)(4n + 1)} \]