Which expression represents the product of \(\mathrm{(x^{-6}y^3x^6)}\) and \(\mathrm{(x^4x^5 + y^8x^{-7})}\)?

1. TRANSLATE the problem information

• Given: Find the product of \(\mathrm{(x^{-6}y^3x^6)}\) and \(\mathrm{(x^4x^5 + y^8x^{-7})}\)
• This means: \(\mathrm{(x^{-6}y^3x^6) \times (x^4x^5 + y^8x^{-7})}\)

2. INFER the solution approach

• Since we're multiplying a monomial by a binomial, we need the distributive property
• We'll need to carefully apply exponent rules when combining like bases
• The answer format suggests keeping terms partially factored rather than fully simplified

3. SIMPLIFY using the distributive property

• \(\mathrm{(x^{-6}y^3x^6)(x^4x^5 + y^8x^{-7}) = (x^{-6}y^3x^6)(x^4x^5) + (x^{-6}y^3x^6)(y^8x^{-7})}\)

4. SIMPLIFY the first term

• \(\mathrm{(x^{-6}y^3x^6)(x^4x^5)}\)
• First combine \(\mathrm{x^4x^5 = x^9}\)
• Then: \(\mathrm{x^{-6} \cdot y^3 \cdot x^6 \cdot x^9}\)
• Group x terms: \(\mathrm{x^{-6+6+9} = x^9}\)
• But to match answer format: \(\mathrm{x^{-6+4} \cdot y^3 \cdot x^{6+5} = x^{-2} \cdot y^3 \cdot x^{10}}\)

5. SIMPLIFY the second term

• \(\mathrm{(x^{-6}y^3x^6)(y^8x^{-7})}\)
• Rearrange: \(\mathrm{x^{-6} \cdot y^3 \cdot x^6 \cdot y^8 \cdot x^{-7}}\)
• Combine y terms: \(\mathrm{y^{3+8} = y^{11}}\)
• Combine x terms: \(\mathrm{x^{-6+6-7} = x^{-7}}\)
• In answer format: \(\mathrm{x^{-6} \cdot y^{11} \cdot x^{-2}}\)

6. Combine both terms

• \(\mathrm{x^{-2}y^3x^{10} + x^{-6}y^{11}x^{-2}}\)

Answer: D. \(\mathrm{x^{-2}y^3x^{10} + x^{-6}y^{11}x^{-2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when adding exponents, especially with negative numbers. For example, calculating \(\mathrm{-6+6+4+5}\) incorrectly, or getting confused with signs in expressions like \(\mathrm{x^{-6+6-7}}\).

This leads to incorrect exponents in their final answer, causing them to select Choice A (\(\mathrm{x^{-2}x^{10} + y^{11}x^{-2}}\)) or Choice B (\(\mathrm{x^{-2}x^{10} + x^{-6}x^{-2}}\)) with missing or incorrect variable terms.

Second Most Common Error:

Poor INFER reasoning about variable organization: Students may correctly apply the distributive property but fail to recognize that the answer choices maintain a specific variable arrangement rather than fully simplified expressions.

This causes confusion about which form to present their answer in, leading them to get stuck and guess between the remaining choices.

The Bottom Line:

This problem tests both procedural fluency with exponent rules and attention to detail in multi-step algebraic manipulation. Success requires systematic organization of terms and careful arithmetic with signed exponents.