(4) \( \frac{\left(3^{2} \cdot 5\right)^{-n+1} \cdot 5^{n-1} \cdot\left(3^{4}\right)^{-1}}{\left(2^{2}\right)^{n+2} \cdot\left(3^{2} \cdot 2^{2}\right)^{n-1}} \)

Let's simplify the expression step by step: \[ \frac{\left(3^{2} \cdot 5\right)^{-n+1} \cdot 5^{n-1} \cdot \left(3^{4}\right)^{-1}}{\left(2^{2}\right)^{n+2} \cdot \left(3^{2} \cdot 2^{2}\right)^{n-1}} \] ### Step 1: Simplify the Numerator 1. **Expand \(\left(3^{2} \cdot 5\right)^{-n+1}\):** \[ \left(3^{2} \cdot 5\right)^{-n+1} = 3^{2(-n+1)} \cdot 5^{-n+1} = 3^{-2n+2} \cdot 5^{-n+1} \] 2. **Multiply by \(5^{n-1}\):** \[ 3^{-2n+2} \cdot 5^{-n+1} \cdot 5^{n-1} = 3^{-2n+2} \cdot 5^{(-n+1) + (n-1)} = 3^{-2n+2} \cdot 5^{0} = 3^{-2n+2} \] 3. **Multiply by \(\left(3^{4}\right)^{-1}\):** \[ 3^{-2n+2} \cdot 3^{-4} = 3^{-2n+2-4} = 3^{-2n-2} \] So, the simplified **numerator** is: \[ 3^{-2n-2} \] ### Step 2: Simplify the Denominator 1. **Expand \(\left(2^{2}\right)^{n+2}\):** \[ \left(2^{2}\right)^{n+2} = 2^{2(n+2)} = 2^{2n+4} \] 2. **Expand \(\left(3^{2} \cdot 2^{2}\right)^{n-1}\):** \[ \left(3^{2} \cdot 2^{2}\right)^{n-1} = 3^{2(n-1)} \cdot 2^{2(n-1)} = 3^{2n-2} \cdot 2^{2n-2} \] 3. **Combine the terms:** \[ 2^{2n+4} \cdot 3^{2n-2} \cdot 2^{2n-2} = 3^{2n-2} \cdot 2^{(2n+4) + (2n-2)} = 3^{2n-2} \cdot 2^{4n+2} \] So, the simplified **denominator** is: \[ 3^{2n-2} \cdot 2^{4n+2} \] ### Step 3: Combine Numerator and Denominator \[ \frac{3^{-2n-2}}{3^{2n-2} \cdot 2^{4n+2}}} \] Simplify the \(3\)-terms: \[ 3^{-2n-2} \div 3^{2n-2} = 3^{-2n-2 - (2n-2)} = 3^{-4n} \] So, the expression becomes: \[ \frac{3^{-4n}}{2^{4n+2}} = \frac{1}{3^{4n} \cdot 2^{4n+2}} \] ### Final Simplified Form \[ \boxed{\dfrac{1}{\,3^{4n}\, \cdot\, 2^{4n + 2}\,}} \] This is the most simplified form of the given expression.