\( \frac { 5 ^ { n - 1 } \cdot 2 ^ { n + 2 } } { 10 ^ { n } - 10 ^ { n - 1 } \cdot 2 } \)

Let's simplify the given expression step by step: \[ \frac{5^{n-1} \cdot 2^{n+2}}{10^{n} - 10^{n-1} \cdot 2} \] ### Step 1: Express \(10^n\) in terms of \(2\) and \(5\) Since \(10 = 2 \times 5\), we can rewrite \(10^n\) as: \[ 10^n = (2 \times 5)^n = 2^n \cdot 5^n \] ### Step 2: Simplify the Denominator Let's simplify the denominator: \[ 10^{n} - 10^{n-1} \cdot 2 = 10^{n-1} \cdot 10 - 10^{n-1} \cdot 2 = 10^{n-1} \cdot (10 - 2) = 10^{n-1} \cdot 8 \] Now, express \(10^{n-1}\) in terms of \(2\) and \(5\): \[ 10^{n-1} = (2 \times 5)^{n-1} = 2^{n-1} \cdot 5^{n-1} \] So, the denominator becomes: \[ 10^{n-1} \cdot 8 = 2^{n-1} \cdot 5^{n-1} \cdot 8 = 2^{n-1} \cdot 5^{n-1} \cdot 2^3 = 2^{n+2} \cdot 5^{n-1} \] ### Step 3: Simplify the Entire Expression Now, plug the simplified numerator and denominator back into the original expression: \[ \frac{5^{n-1} \cdot 2^{n+2}}{2^{n+2} \cdot 5^{n-1}} = \frac{5^{n-1}}{5^{n-1}} \cdot \frac{2^{n+2}}{2^{n+2}} = 1 \cdot 1 = 1 \] ### Conclusion The expression simplifies to: \[ \boxed{1} \] No matter the value of \(n\) (provided it's within the domain where the expression is defined), the expression always equals **1**.