Step 1: Write the given rate law.
\[ Rate = k[A][B]^{3/2} \]
Step 2: Calculate the overall order.
Overall order = sum of all exponents in the rate law: \[ \text{Order} = 1 + \frac{3}{2} = \frac{5}{2} = 2.5 \]
Step 3: Define what an elementary reaction requires.
An elementary reaction occurs in a single mechanistic step exactly as written. For such a reaction: (i) Each exponent in the rate law must equal the stoichiometric coefficient of that species. (ii) All exponents must be whole positive integers. (iii) The molecularity (= overall order for elementary reactions) must be 1, 2, or rarely 3.
Step 4: Identify why this reaction cannot be elementary.
The order with respect to B is $3/2$, a fraction. Molecularity cannot be fractional (you cannot have half a molecule participating in a single elementary step). Fractional orders arise only in complex (multi-step) mechanisms where the rate-determining step involves concentration terms derived from equilibrium steps before it.
Step 5: Confirm the reasoning.
Only a complex mechanism involving multiple steps can give rise to a fractional order of $3/2$ for B. This fractional power is the mathematical fingerprint of a multi-step mechanism, not a single elementary step.
Step 6: State both answers clearly.
Overall order = 2.5. The reaction CANNOT be elementary because the order is fractional (2.5), and elementary reactions must have integral (whole-number) orders equal to the molecularity. \[ \boxed{\text{Order} = 2.5;\; \text{NOT elementary (fractional order impossible for elementary reactions)}} \]