Which expression is equivalent to (x^6 y^2 w^(-3))/(x^2 y^(-1) w^(-5)), where x, y, and w are positive?

1. TRANSLATE the problem information

• Given: \(\mathrm{x}^6 \mathrm{y}^2 \mathrm{w}^{-3} \div \mathrm{x}^2 \mathrm{y}^{-1} \mathrm{w}^{-5}\)
• Need to find: Equivalent simplified expression

2. INFER the solution approach

• Since we're dividing expressions with the same bases (x, y, w), we need to apply the exponent division rule to each variable separately
• Key insight: When dividing powers with the same base, subtract the exponents

3. SIMPLIFY each variable using the division rule \(\mathrm{a}^m \div \mathrm{a}^n = \mathrm{a}^{m-n}\)

• For x: \(\mathrm{x}^6 \div \mathrm{x}^2 = \mathrm{x}^{6-2} = \mathrm{x}^4\)

• For y: \(\mathrm{y}^2 \div \mathrm{y}^{-1} = \mathrm{y}^{2-(-1)} = \mathrm{y}^{2+1} = \mathrm{y}^3\)
  (Be careful: subtracting a negative means adding!)

• For w: \(\mathrm{w}^{-3} \div \mathrm{w}^{-5} = \mathrm{w}^{-3-(-5)} = \mathrm{w}^{-3+5} = \mathrm{w}^2\)
  (Again, subtracting a negative becomes addition)

4. Combine the results

• Final expression: \(\mathrm{x}^4 \mathrm{y}^3 \mathrm{w}^2\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative exponents: Students forget that subtracting a negative exponent means adding the absolute value.

For example, they might calculate:

• \(\mathrm{y}^2 \div \mathrm{y}^{-1} = \mathrm{y}^{2-1} = \mathrm{y}^1\) (forgetting to handle the negative sign properly)
• \(\mathrm{w}^{-3} \div \mathrm{w}^{-5} = \mathrm{w}^{-3-5} = \mathrm{w}^{-8}\) (adding the exponents instead of subtracting)

This combination of errors may lead them to select Choice C (\(\mathrm{x}^8 \mathrm{y}^1 \mathrm{w}^{-8}\)) or another incorrect option.

Second Most Common Error:

Conceptual confusion about division vs. multiplication rules: Students might add exponents instead of subtracting them, confusing the division rule (\(\mathrm{a}^m \div \mathrm{a}^n = \mathrm{a}^{m-n}\)) with the multiplication rule (\(\mathrm{a}^m \times \mathrm{a}^n = \mathrm{a}^{m+n}\)).

This leads to calculations like \(\mathrm{x}^6 \div \mathrm{x}^2 = \mathrm{x}^{6+2} = \mathrm{x}^8\), producing incorrect exponents that don't match any answer choice, causing confusion and guessing.

The Bottom Line:

This problem tests your precision with exponent rules, especially when negative exponents are involved. The key is remembering that "subtract the exponent" means you must carefully handle negative signs: subtracting a negative becomes addition.