Question:Let a, b, and c be positive real numbers. Which expression is equivalent to \(\mathrm{(a^3b^{-2}c^4)^2 \div (a^{-1}b^3c)}\)?a^7b^(-7)c^7a^7b^...

1. TRANSLATE the problem information

• Given expression: \(\mathrm{a}^3\mathrm{b}^{-2}\mathrm{c}^4)^2 \div (\mathrm{a}^{-1}\mathrm{b}^3\mathrm{c})\)
• Need to: Simplify this expression to match one of the answer choices

2. INFER the correct approach

• Order of operations matters: Handle the power first, then the division
• For division of expressions with same bases, subtract exponents
• Work systematically through each variable (a, b, c)

3. SIMPLIFY by applying the power rule

• Square the numerator: \(\mathrm{a}^3\mathrm{b}^{-2}\mathrm{c}^4)^2\)
• Apply power rule \(\mathrm{a}^\mathrm{m})^\mathrm{n} = \mathrm{a}^{\mathrm{mn}}\) to each factor:
  • \(\mathrm{a}^3\) squared: \(\mathrm{a}^{3 \times 2} = \mathrm{a}^6\)
  • \(\mathrm{b}^{-2}\) squared: \(\mathrm{b}^{-2 \times 2} = \mathrm{b}^{-4}\)
  • \(\mathrm{c}^4\) squared: \(\mathrm{c}^{4 \times 2} = \mathrm{c}^8\)
• Result: \(\mathrm{a}^6\mathrm{b}^{-4}\mathrm{c}^8\)

4. SIMPLIFY by applying the division rule

• Now divide: \(\mathrm{a}^6\mathrm{b}^{-4}\mathrm{c}^8) \div (\mathrm{a}^{-1}\mathrm{b}^3\mathrm{c})\)
• For each variable, subtract the denominator exponent from numerator exponent:
  • For a: \(6 - (-1) = 6 + 1 = 7\) → \(\mathrm{a}^7\)
  • For b: \(-4 - 3 = -7\) → \(\mathrm{b}^{-7}\)
  • For c: \(8 - 1 = 7\) → \(\mathrm{c}^7\)

Answer: (A) \(\mathrm{a}^7\mathrm{b}^{-7}\mathrm{c}^7\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when working with negative exponents, particularly when subtracting a negative exponent.

For the 'a' term: Instead of calculating \(6 - (-1) = 7\), they might compute \(6 - 1 = 5\), forgetting that subtracting a negative means adding. This type of error, combined with other calculation mistakes, leads to confusion about which answer choice is correct and often results in guessing.

Second Most Common Error:

Poor INFER reasoning about order of operations: Students attempt to distribute the division before applying the power rule, trying to simplify \(\mathrm{a}^3\mathrm{b}^{-2}\mathrm{c}^4) \div (\mathrm{a}^{-1}\mathrm{b}^3\mathrm{c})\) first, then squaring the result.

This incorrect approach makes the problem much more complicated and typically leads to arithmetic errors. Students may abandon their systematic approach and select Choice (C) \(\mathrm{a}^6\mathrm{b}^{-4}\mathrm{c}^8)\) because it looks like the result of just applying the power rule without the division.

The Bottom Line:

This problem tests whether students can systematically apply multiple exponent rules while carefully tracking positive and negative signs. The key is working step-by-step and being extra careful with the arithmetic when negative exponents are involved.