Given that $\frac{5^{n +3}}{25^{2n -3}}$ = 5º, find n

Analyze the question: We need n such that 5^(n+3) ÷ 25^(2n−3) = 5⁰ = 1.

Concept: Express everything with the same base, then use the law of indices: aᵐ ÷ aᵖ = aᵐ⁻ᵖ. If the bases are equal, the powers must be equal.

Solution:

1. Write 25 as a power of 5: 25 = 5², so 25^(2n−3) = 5^(2(2n−3)) = 5^(4n−6).

2. Substitute: 5^(n+3) ÷ 5^(4n−6) = 5^((n+3) − (4n−6)) = 5^(−3n+9)

3. Equate to 5⁰: −3n + 9 = 0 3n = 9 n = 3

Check: With n = 3, 5^(6) ÷ 25^(3) = 5⁶ ÷ 5⁶ = 1 ✓

The correct answer is **C.

Tip: When you see mixed bases like 5 and 25, convert to a common base first (25 = 5², 8 = 2³, 9 = 3²). Then just equate the exponents. Also remember that anything raised to the power 0 equals 1, so the exponent must be 0.