Which expression is equivalent to \(\mathrm{m}^4\mathrm{q}^4\mathrm{z}^{-1})mq^3z^3, where m, q, and z are positive?

1. TRANSLATE the problem information

• Given expression: \(\mathrm{m}^4\mathrm{q}^4\mathrm{z}^{-1})\(\mathrm{m}\mathrm{q}^3\mathrm{z}^3\)
• We need to multiply two algebraic expressions containing variables with exponents

2. INFER the approach

• This is an exponent multiplication problem where we have the same bases (m, q, z) in both expressions
• Strategy: Group like terms and apply the rule that \(\mathrm{x}^\mathrm{a} \times \mathrm{x}^\mathrm{b} = \mathrm{x}^{\mathrm{a}+\mathrm{b}}\)
• We can mentally regroup as \(\mathrm{m}^4 \times \mathrm{m})\(\mathrm{q}^4 \times \mathrm{q}^3)\(\mathrm{z}^{-1} \times \mathrm{z}^3\)

3. SIMPLIFY by applying exponent rules to each variable

• For m: \(\mathrm{m}^4 \times \mathrm{m}^1 = \mathrm{m}^{4+1} = \mathrm{m}^5\)
• For q: \(\mathrm{q}^4 \times \mathrm{q}^3 = \mathrm{q}^{4+3} = \mathrm{q}^7\)
• For z: \(\mathrm{z}^{-1} \times \mathrm{z}^3 = \mathrm{z}^{-1+3} = \mathrm{z}^2\)

4. Combine the results

• Final expression: \(\mathrm{m}^5\mathrm{q}^7\mathrm{z}^2\)

Answer: B. \(\mathrm{m}^5\mathrm{q}^7\mathrm{z}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Multiplying exponents instead of adding them

Students often confuse the rule \(\mathrm{x}^\mathrm{a})^\mathrm{b} = \mathrm{x}^{\mathrm{ab}}\) with the rule \(\mathrm{x}^\mathrm{a} \times \mathrm{x}^\mathrm{b} = \mathrm{x}^{\mathrm{a}+\mathrm{b}}\). They might calculate:

• m: \(4 \times 1 = 4 \rightarrow \mathrm{m}^4\)
• q: \(4 \times 3 = 12 \rightarrow \mathrm{q}^{12}\)
• z: \((-1) \times 3 = -3 \rightarrow \mathrm{z}^{-3}\)

This may lead them to select Choice A (\(\mathrm{m}^4\mathrm{q}^{20}\mathrm{z}^{-3}\)) if they also make calculation errors, or create confusion leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors with negative exponents

Students correctly identify that they need to add exponents but make mistakes with negative numbers. For \(\mathrm{z}^{-1} + 3\), they might calculate incorrectly as -4 or -1 instead of +2.

This leads to selecting incorrect answer choices or abandoning systematic solution and guessing.

The Bottom Line:

This problem tests whether students can correctly apply the fundamental exponent rule for multiplication while carefully handling negative exponents and multi-step arithmetic.