100^n/ 5^2n-1 × 4^n-1

To simplify the expression \( \frac{100^n}{5^{2n-1} \times 4^{n-1}} \), let's break it down step by step: 1. **Express 100 and 4 in terms of their prime factors**: \[ 100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \] \[ 4 = 2^2 \] 2. **Rewriting the expression**: Substitute \( 100^n \) and \( 4^{n-1} \) in the original expression: \[ 100^n = (2^2 \times 5^2)^n = 2^{2n} \times 5^{2n} \] \[ 4^{n-1} = (2^2)^{n-1} = 2^{2(n-1)} = 2^{2n-2} \] 3. **Now substitute these back into the expression**: \[ \frac{100^n}{5^{2n-1} \times 4^{n-1}} = \frac{2^{2n} \times 5^{2n}}{5^{2n-1} \times 2^{2n-2}} \] 4. **Combine the terms**: The numerator is \( 2^{2n} \times 5^{2n} \), and the denominator is \( 5^{2n-1} \times 2^{2n-2} \). We can simplify: \[ = \frac{2^{2n}}{2^{2n-2}} \times \frac{5^{2n}}{5^{2n-1}} = 2^{2n - (2n - 2)} \times 5^{2n - (2n - 1)} = 2^2 \times 5^1 \] 5. **Final simplification**: \[ = 4 \times 5 = 20 \] Thus, the final simplified result is \( \boxed{20} \).