Which expression is equivalent to 42a/k + 42ak, where k gt 0?

1. TRANSLATE the problem information

• Given: \(\frac{42a}{k} + 42ak\), where \(k \gt 0\)
• Need: Equivalent expression from the choices

2. INFER the approach needed

• These are two terms that need to be added
• The first term \(\frac{42a}{k}\) is a fraction with denominator k
• The second term \(42ak\) has an implicit denominator of 1
• To add fractions, I need a common denominator

3. SIMPLIFY by creating a common denominator

• Multiply the second term by \(\frac{k}{k}\) (which equals 1, so doesn't change the value):
  \(42ak \times \frac{k}{k} = \frac{42ak²}{k}\)
• Now both terms have denominator k

4. SIMPLIFY by adding the fractions

• \(\frac{42a}{k} + \frac{42ak²}{k} = \frac{42a + 42ak²}{k}\)
• Factor out the common factor 42a from the numerator:
  \(\frac{42a(1 + k²)}{k} = \frac{42a(k² + 1)}{k}\)

Answer: D. \(\frac{42a(k²+1)}{k}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that fractions need a common denominator before adding. Instead, they might try to add the numerators and denominators separately or combine terms incorrectly.

This leads them to select Choice A \(\frac{84a}{k}\) by incorrectly thinking \(\frac{42a}{k} + 42ak = \frac{84a}{k}\), treating both terms as if they have the same denominator.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the need for a common denominator but make algebraic errors when multiplying \(42ak\) by \(\frac{k}{k}\), perhaps getting \(\frac{42ak}{k}\) instead of \(\frac{42ak²}{k}\).

This leads to incorrect intermediate steps and confusion, causing them to guess among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically work with algebraic fractions, requiring both strategic thinking about common denominators and careful algebraic manipulation through multiple steps.