If p and q are positive numbers, which of the following is equivalent to \(\mathrm{(p + q)^{5/3} \cdot (p +...

1. SIMPLIFY using exponent rules

• Given: \((\mathrm{p} + \mathrm{q})^{5/3} \cdot (\mathrm{p} + \mathrm{q})^{1/3}\)
• Since we're multiplying powers with the same base \((\mathrm{p} + \mathrm{q})\), we add the exponents:
  • \(\frac{5}{3} + \frac{1}{3} = \frac{6}{3} = 2\)
  • So we get: \((\mathrm{p} + \mathrm{q})^2\)

2. SIMPLIFY by expanding the perfect square

• We need to expand \((\mathrm{p} + \mathrm{q})^2\) using the perfect square formula
• \((\mathrm{p} + \mathrm{q})^2 = \mathrm{p}^2 + 2\mathrm{pq} + \mathrm{q}^2\)

Answer: (C) p² + 2pq + q²

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with fraction arithmetic: Students correctly identify they need to add exponents but make errors with the fraction addition, calculating \(\frac{5}{3} + \frac{1}{3}\) incorrectly or getting confused about simplifying \(\frac{6}{3}\).

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge about perfect square expansion: Students successfully get to \((\mathrm{p} + \mathrm{q})^2\) but then expand it incorrectly as just \(\mathrm{p}^2 + \mathrm{q}^2\), forgetting the middle term \(2\mathrm{pq}\).

This may lead them to select Choice (B) (p² + q²).

Third Most Common Error:

Weak SIMPLIFY skill with exponent rules: Students confuse the multiplication rule with other exponent operations and multiply the exponents instead of adding them, getting \((\mathrm{p} + \mathrm{q})^{5/9}\) or attempt some other incorrect manipulation.

This leads to confusion and guessing since none of the answer choices match their incorrect intermediate result.

The Bottom Line:

This problem tests two fundamental algebraic skills in sequence - students must correctly apply exponent rules AND remember binomial expansion. Success requires solid execution of both computational steps without error.