Question:An expression is formed by dividing p^4q^(-2) + p^(-3)q^5 by p^(-5)q^2p^5q^1. Which of the following is an equivalent expression?p^6q^(-5)p^(...

1. TRANSLATE the problem information

• Given information:
  • Numerator: \(\mathrm{p^4q^{-2} + p^{-3}q^5}\)
  • Divisor: \(\mathrm{p^{-5}q^2p^5q^1}\)
  • Need to find the equivalent expression after division

2. INFER the most efficient approach

• The divisor has multiple terms with the same base - combine these first
• Then distribute the division to each term in the numerator
• This approach prevents working with overly complex fractions

3. SIMPLIFY the divisor by combining like terms

• In \(\mathrm{p^{-5}q^2p^5q^1}\), group terms with same bases:
  • p terms: \(\mathrm{p^{-5} \cdot p^5 = p^{-5+5} = p^0 = 1}\)
  • q terms: \(\mathrm{q^2 \cdot q^1 = q^{2+1} = q^3}\)
• Simplified divisor: \(\mathrm{1 \cdot q^3 = q^3}\)

4. SIMPLIFY the division by distributing to each term

• The expression becomes: \(\mathrm{(p^4q^{-2} + p^{-3}q^5) \div q^3}\)
• Distribute division: \(\mathrm{(p^4q^{-2}) \div q^3 + (p^{-3}q^5) \div q^3}\)

5. SIMPLIFY each term using division rules for exponents

• First term: \(\mathrm{p^4q^{-2} \div q^3}\)
  • Since dividing q terms: \(\mathrm{q^{-2} \div q^3 = q^{-2-3} = q^{-5}}\)
  • Result: \(\mathrm{p^4q^{-5}}\)
• Second term: \(\mathrm{p^{-3}q^5 \div q^3}\)
  • Since dividing q terms: \(\mathrm{q^5 \div q^3 = q^{5-3} = q^2}\)
  • Result: \(\mathrm{p^{-3}q^2}\)
• Combined result: \(\mathrm{p^4q^{-5} + p^{-3}q^2}\)

6. INFER which answer choice matches by testing

• Check option (B): \(\mathrm{p^6q^{-5}p^{-2} + p^{-1}q^2p^{-2}}\)
  • First term: \(\mathrm{p^6 \cdot p^{-2} \cdot q^{-5} = p^4q^{-5}}\) ✓
  • Second term: \(\mathrm{p^{-1} \cdot p^{-2} \cdot q^2 = p^{-3}q^2}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when working with negative exponents, especially in the subtraction step of division rules.

For example, when computing \(\mathrm{q^{-2} \div q^3}\), they might calculate \(\mathrm{q^{-2-3} = q^{-5}}\) as \(\mathrm{q^1}\) or \(\mathrm{q^{-1}}\) instead. Similarly, they might compute \(\mathrm{q^5 \div q^3 = q^{5-3} = q^2}\) incorrectly as \(\mathrm{q^8}\) (adding instead of subtracting). These calculation errors directly lead them to select Choice (A) or Choice (D) depending on which specific mistakes they make.

Second Most Common Error:

Poor INFER reasoning about problem structure: Students attempt to divide each term in the numerator by the entire unsimplified divisor \(\mathrm{p^{-5}q^2p^5q^1}\), creating unnecessarily complex fractions.

This approach becomes algebraically messy and prone to errors. Students often abandon systematic work partway through and resort to guessing among the answer choices, or make computational mistakes that lead them away from the correct answer.

The Bottom Line:

This problem tests whether students can systematically apply exponent rules while maintaining accuracy with negative exponents. The key insight is recognizing that simplifying the divisor first makes the entire problem much more manageable than trying to work with the complex original form.