Let a, b, and c be positive numbers. Which of the following expressions is equivalent to the expression below?(a^(6)b^(3)c^(-4))/(a^(2)b^(-5)c^(-7))

1. INFER the solution strategy

• Recognize this is a division of exponential expressions
• The quotient rule for exponents applies: \(\mathrm{x^m / x^n = x^{(m-n)}}\)
• Apply this rule separately to each variable (a, b, and c)

2. SIMPLIFY the expression for variable 'a'

• For \(\mathrm{a^6 / a^2}\), subtract exponents: \(\mathrm{6 - 2 = 4}\)
• Result: \(\mathrm{a^4}\)

3. SIMPLIFY the expression for variable 'b'

• For \(\mathrm{b^3 / b^{-5}}\), subtract exponents: \(\mathrm{3 - (-5) = 3 + 5 = 8}\)
• Key point: Subtracting a negative is the same as adding
• Result: \(\mathrm{b^8}\)

4. SIMPLIFY the expression for variable 'c'

• For \(\mathrm{c^{-4} / c^{-7}}\), subtract exponents: \(\mathrm{-4 - (-7) = -4 + 7 = 3}\)
• Again, subtracting a negative becomes addition
• Result: \(\mathrm{c^3}\)

5. Combine all results

• Final answer: \(\mathrm{a^4b^8c^3}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when working with negative exponents

Students often struggle with the arithmetic: "3 minus negative 5." They might incorrectly calculate \(\mathrm{b^3 / b^{-5}}\) as \(\mathrm{b^{3-5} = b^{-2}}\), instead of recognizing that \(\mathrm{3 - (-5) = 3 + 5 = 8}\). Similarly, they might get \(\mathrm{c^{-4} / c^{-7}}\) wrong by doing \(\mathrm{-4 - 7 = -11}\) instead of \(\mathrm{-4 - (-7) = 3}\).

This may lead them to select Choice A (\(\mathrm{a^4b^{-2}c^{-11}}\)) or Choice D (\(\mathrm{a^4b^{-8}c^{-3}}\)).

Second Most Common Error:

Poor INFER reasoning: Misapplying or forgetting the quotient rule

Some students might try to multiply exponents instead of subtracting them, or they might add exponents instead of subtracting. Others might only apply the rule to some variables and not others, leading to partial solutions.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests both conceptual knowledge of exponent rules and careful arithmetic with negative numbers. The key insight is that "subtracting a negative" always becomes "adding a positive."