Which expression is equivalent to 13a^2 - 7a^2?

1. INFER what type of expression we have

• Given: \(\mathrm{13a^2 - 7a^2}\)
• Both terms contain \(\mathrm{a^2}\) as the variable part
• Since both terms have identical variable parts, they are like terms
• Like terms can be combined by adding or subtracting their coefficients

2. SIMPLIFY by combining the like terms

• Method 1 (Factor out common factor): \(\mathrm{13a^2 - 7a^2 = (13 - 7)a^2 = 6a^2}\)
• Method 2 (Direct combination): \(\mathrm{13a^2 - 7a^2 = 6a^2}\)
• Either way, we subtract the coefficients: \(\mathrm{13 - 7 = 6}\)

Answer: B. \(\mathrm{6a^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that these are like terms that should be combined

Students might think they need to multiply the terms instead of combine them, leading to calculations like \(\mathrm{13 \times (-7) = -91}\), causing them to select Choice A (\(\mathrm{-91a^2}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Performing the wrong arithmetic operation

Instead of subtracting \(\mathrm{(13 - 7)}\), students might add the coefficients \(\mathrm{(13 + 7 = 20)}\), leading them to select Choice C (\(\mathrm{20a^2}\)).

The Bottom Line:

This problem tests whether students can identify like terms and apply the correct operation. The key insight is recognizing that when terms have the same variable part, you combine their coefficients using the given operation (subtraction in this case).