Let p and q be positive numbers. Which of the following is equivalent to \((\mathrm{p} + \mathrm{q})^{3/2} \div \sqrt{\mathrm{p} +...

1. TRANSLATE the radical to exponential form

• Given expression: \((p + q)^{\frac{3}{2}} \div \sqrt{p + q}\)
• Convert the square root: \(\sqrt{p + q} = (p + q)^{\frac{1}{2}}\)
• Rewritten expression: \((p + q)^{\frac{3}{2}} \div (p + q)^{\frac{1}{2}}\)

2. SIMPLIFY using the law of exponents

• When dividing powers with the same base, subtract the exponents:
  \((p + q)^{\frac{3}{2}} \div (p + q)^{\frac{1}{2}} = (p + q)^{(\frac{3}{2} - \frac{1}{2})}\)
• Subtract the fractions: \(\frac{3}{2} - \frac{1}{2} = \frac{2}{2} = 1\)
• Therefore: \((p + q)^1 = p + q\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not recognizing that \(\sqrt{p + q}\) can be written as \((p + q)^{\frac{1}{2}}\)

Students often try to work with the radical form directly, attempting to factor or expand \((p + q)^{\frac{3}{2}}\) instead of converting everything to exponential form. Without this key translation, they can't apply the law of exponents and may resort to guessing or selecting a more complicated-looking answer.

This may lead them to select Choice C (\(p^2 + 2pq + q^2\)) thinking they need to expand something, or get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when subtracting fractional exponents

Students correctly convert to \((p + q)^{\frac{3}{2}} \div (p + q)^{\frac{1}{2}}\) but then incorrectly calculate \(\frac{3}{2} - \frac{1}{2}\). Some might get confused and think it equals \(\frac{3}{2} - \frac{1}{2} = \frac{2}{1} = 2\), leading them to \((p + q)^2\).

This may lead them to select Choice C (\(p^2 + 2pq + q^2\)) since \((p + q)^2 = p^2 + 2pq + q^2\).

The Bottom Line:

This problem tests whether students can recognize when to convert between radical and exponential notation. The key insight is that once everything is in exponential form with the same base, the division becomes straightforward using the law of exponents.