What is the value of 3² × 2³?

The expression 3² × 2³ evaluates to 72 when each exponential term is computed separately according to the definition of exponentiation (repeated multiplication), yielding 3² = 9 and 2³ = 8, then multiplying these results together to obtain 9 × 8 = 72 as the final value.

A. 36

This value equals 6² or 4 × 9, suggesting students likely added the exponents (2 + 3 = 5) then incorrectly computed 3⁵ = 243 or 2⁵ = 32 before approximating downward, or more plausibly executed 3² = 9 and 2³ = 8 correctly but then added rather than multiplied the results (9 + 8 = 17) then adjusted upward to 36 through unstructured guesswork. Another error pathway involves misapplying exponent rules to multiply bases first (3 × 2 = 6) then adding exponents (2 + 3 = 5) to get 6⁵ = 7,776 then recognizing absurdity and guessing 36 as a "reasonable" alternative. The value 36 also equals (3 × 2)² = 6², revealing possible confusion between (ab)ⁿ = aⁿbⁿ versus aⁿ × bᵐ where n ≠ m a critical exponent rule distinction where identical exponents permit base multiplication but different exponents require separate evaluation. This distractor effectively identifies students who conflate exponent operations with arithmetic operations on bases/exponents or who lack procedural sequencing clarity in multi-operation expressions requiring strict order of operations adherence.

B. 48

This amount equals 6 × 8 or 16 × 3, suggesting students likely computed 2³ = 8 correctly but then miscalculated 3² as 6 (confusing squaring with doubling: 3 × 2 = 6 instead of 3 × 3 = 9) before multiplying 6 × 8 = 48. This "doubling versus squaring" error represents a fundamental misconception about exponentiation where learners interpret a² as 2a rather than a × a a pervasive deficiency particularly common among students newly introduced to exponential notation who have not yet internalized that exponents represent repeated multiplication rather than scalar multiplication. Another plausible pathway involves computing 3² = 9 and 2³ = 8 correctly but then executing 9 × 8 = 72 then erroneously subtracting 24 (perhaps confusing with another problem) to reach 48. The 24-unit deficit (72 - 48 = 24) equals precisely 3 × 8, suggesting possible term omission during multiplication. This distractor proves particularly instructive for diagnosing conceptual misunderstandings of exponentiation itself rather than computational errors a foundational deficiency requiring explicit reinforcement of exponent definition through concrete repeated multiplication demonstrations (3² = 3 × 3, not 3 × 2) before advancing to exponent rules and complex expressions.

C. 72

This value correctly represents the expression evaluation through multiple verification pathways demonstrating computational fluency and conceptual understanding. Primary evaluation: 3² = 3 × 3 = 9; 2³ = 2 × 2 × 2 = 8; 9 × 8 = 72. Prime factorization verification: 3² × 2³ = 9 × 8 = (3 × 3) × (2 × 2 × 2) = 72; rearranging factors: 2 × 2 × 2 × 3 × 3 = 8 × 9 = 72. Exponent rule application: recognizing no simplification rules apply across different bases with different exponents, requiring separate evaluation then multiplication. Calculator verification: entering 3^2*2^3 yields 72. This solution demonstrates mastery of exponentiation fundamentals including definition comprehension (repeated multiplication), order of operations sequencing (exponents before multiplication), multi-step computation management, and verification through factor rearrangement integrated competencies essential for algebraic simplification, scientific notation manipulation, exponential growth modeling, and computational mathematics where precise exponent evaluation forms the foundation for increasingly sophisticated quantitative reasoning.

D. 96

This figure equals 12 × 8 or 32 × 3, suggesting students likely computed 2³ = 8 correctly but then miscalculated 3² as 12 (perhaps confusing with 3! = 6 then doubling, or adding 3 + 3 + 3 + 3 = 12 misinterpreting exponent as repeated addition count rather than multiplier count). Another plausible error pathway involves computing 3² = 9 and 2³ = 8 correctly but then multiplying 9 × 8 = 72 then erroneously adding 24 (72 + 24 = 96) through unstructured adjustment or misreading the operation as addition after multiplication. The 24-unit excess (96 - 72 = 24) equals precisely 3 × 8, suggesting possible double-counting of a factor during multiplication. This distractor effectively targets learners who conflate exponentiation with factorial operations or repeated addition patterns a transitional misconception requiring explicit differentiation between operation types: exponents (repeated multiplication), factorials (descending multiplication), and multiplication itself (repeated addition) each with distinct symbolic representations and computational procedures essential for algebraic literacy and advanced mathematical communication.

Conclusion

The value 72 emerges through precise evaluation of each exponential term followed by multiplication, adhering strictly to order of operations principles where exponents are resolved before multiplicative combinations. This problem reinforces foundational exponential literacy essential across mathematical domains: understanding exponentiation as repeated multiplication rather than scalar operations, executing order of operations with precision (PEMDAS/BODMAS), recognizing when exponent rules apply versus when separate evaluation is required, and verifying results through prime factorization or alternative computational pathways. Mastery of these integrated competencies proves indispensable for scientific notation (astronomical distances, microscopic measurements), compound interest calculations (financial growth modeling), algorithmic complexity analysis (computer science), and exponential decay modeling (radioactive half-lives, pharmacokinetics) where accurate exponent evaluation determines solution validity across STEM disciplines. The distractors strategically target pervasive misconceptions including doubling versus squaring confusion, exponent-addition errors, and operation conflation highlighting the necessity of explicit exponent definition reinforcement through concrete repeated multiplication demonstrations before advancing to abstract rules, ensuring learners develop robust conceptual foundations supporting accurate evaluation of increasingly complex exponential expressions.