Which expression is equivalent to 8a^3b^2 + 24a^3b^2? \(8\mathrm{a}^3\mathrm{b}^2(8)\) \(8\mathrm{a}^3\mathrm{b}^2(24)\) \(8\mathrm{a}^3\mathrm{b}^2...

1. INFER what the problem is asking

• We need to find which expression equals \(8\mathrm{a}^3\mathrm{b}^2 + 24\mathrm{a}^3\mathrm{b}^2\)
• Notice both terms have the exact same variable part: \(\mathrm{a}^3\mathrm{b}^2\)
• This means they are like terms that can be combined

2. SIMPLIFY by combining like terms

• When combining like terms, add the coefficients: \(8\mathrm{a}^3\mathrm{b}^2 + 24\mathrm{a}^3\mathrm{b}^2 = (8 + 24)\mathrm{a}^3\mathrm{b}^2\)
• Calculate: \(8 + 24 = 32\)
• So we get: \(32\mathrm{a}^3\mathrm{b}^2\)

3. SIMPLIFY by checking which answer choice equals \(32\mathrm{a}^3\mathrm{b}^2\)

• (A) \(8\mathrm{a}^3\mathrm{b}^2(8) = 8 \times 8 \times \mathrm{a}^3\mathrm{b}^2 = 64\mathrm{a}^3\mathrm{b}^2\) ❌
• (B) \(8\mathrm{a}^3\mathrm{b}^2(24) = 8 \times 24 \times \mathrm{a}^3\mathrm{b}^2 = 192\mathrm{a}^3\mathrm{b}^2\) ❌
• (C) \(8\mathrm{a}^3\mathrm{b}^2(3) = 8 \times 3 \times \mathrm{a}^3\mathrm{b}^2 = 24\mathrm{a}^3\mathrm{b}^2\) ❌
• (D) \(8\mathrm{a}^3\mathrm{b}^2(4) = 8 \times 4 \times \mathrm{a}^3\mathrm{b}^2 = 32\mathrm{a}^3\mathrm{b}^2\) ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(8\mathrm{a}^3\mathrm{b}^2\) and \(24\mathrm{a}^3\mathrm{b}^2\) are like terms that can be combined.

Instead, they might try to factor each term separately or get confused about what to do with the expression. They may randomly select an answer choice without systematically combining the terms first. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify like terms but make arithmetic errors.

Common mistakes include calculating \(8 + 24 = 30\) (instead of 32) or making errors when evaluating the answer choices (like thinking \(8 \times 4 = 36\)). If they get \(8 + 24 = 30\), they might look for \(30\mathrm{a}^3\mathrm{b}^2\) and not find it among the choices, leading to frustration and guessing.

The Bottom Line:

This problem tests whether students can identify like terms and combine them correctly. The key insight is recognizing that terms with identical variable parts can be combined by adding their coefficients, then checking which factored form represents the same value.