(17)3.5 x (17)? = 178

When two powers share the same base, multiplying them keeps the base and adds the exponents: am x an = am+n. So whenever two powers of the SAME base are equal, their exponents themselves must be equal — if ap = aq (same base a), then p = q.

Applying this to the given equation:

1. Write both sides with base 17: (17)3.5 x (17)x = 178.

2. By the product rule, the left side combines into a single power: (17)3.5 + x.

3. Since both sides now have the same base 17, the exponents must be equal: 3.5 + x = 8.

4. Solve for x: x = 8 − 3.5 = 4.5.

Cross-check: substituting x = 4.5 gives 3.5 + 4.5 = 8, so (17)3.5 x (17)4.5 = 178 — the equation holds exactly.

So the missing exponent is 4.5.