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 \( \boxed{20} \).