Find the value of x, if (16)2.5 × (16)x = (16)5.

Concept: When two powers with the same base are multiplied, add their exponents — for any base a ≠ 0 and exponents m, n: am × an = a(m + n). If two powers of the same base are equal, their exponents must also be equal.

Application:

1. The given equation is (16)2.5 × (16)x = (16)5.

2. Since both terms on the left have the same base 16, apply the product rule to combine them: (16)(2.5 + x) = (16)5.

3. The bases on both sides are equal (16), so the exponents must be equal: 2.5 + x = 5.

4. Isolate x: x = 5 − 2.5 = 2.5.

Cross-check: Substituting x = 2.5 back, the left exponent becomes 2.5 + 2.5 = 5, which matches the right-hand exponent exactly, so (16)2.5 × (16)2.5 = (16)5 holds.

So, the value of x is 2.5.