The value of (5n+3 − 6 × 5n+1) divided by (9 × 5n − 4 × 5n) is

Concept: for a common base a, am+n = am · an. When a sum/difference of exponential terms shares a common base, factor out the lowest power of that base from every term — this turns the expression into a plain numeric ratio, since the remaining base-power factor cancels between numerator and denominator.

Application:

1. Rewrite the numerator using the exponent law: 5n+3 = 5n · 53 and 5n+1 = 5n · 5, so the numerator becomes 5n · 53 − 6 · 5n · 5 = 5n(53 − 6 × 5).

2. Evaluate the bracket: 53 = 125 and 6 × 5 = 30, so the numerator simplifies to 5n(125 − 30) = 95 × 5n.

3. Rewrite the denominator the same way: 9 × 5n − 4 × 5n = (9 − 4) × 5n = 5 × 5n.

4. Divide the simplified numerator by the simplified denominator: (95 × 5n) / (5 × 5n). The 5n factor cancels (it is common and non-zero), leaving 95 / 5 = 19.

Cross-check: substitute a concrete value, say n = 0. Numerator = 53 − 6 × 51 = 125 − 30 = 95. Denominator = 9 × 50 − 4 × 50 = 9 − 4 = 5. Ratio = 95 / 5 = 19 — the same result, confirming the value is independent of n as the algebra predicted.