(25)7.5 x (5)2.5 ÷ (125)1.5 = 5?

Concept: To simplify products and quotients of exponential terms with different bases, first express every base as a power of the same prime base. Then apply the laws of exponents: (am)n = a(mn); am × an = a(m+n); am ÷ an = a(m−n).

1. 25 = 52, so (25)7.5 = (52)7.5 = 515 (since 2 × 7.5 = 15).

2. 125 = 53, so (125)1.5 = (53)1.5 = 54.5 (since 3 × 1.5 = 4.5).

3. Substituting back: 515 × 52.5 ÷ 54.5 = 5x, so the exponents combine as 15 + 2.5 − 4.5.

4. 15 + 2.5 − 4.5 = 13, so 513 = 5x, giving x = 13.

Cross-check: 15 + 2.5 = 17.5, and 17.5 − 4.5 = 13, so the arithmetic is consistent, confirming x = 13.