Let p and q be positive real numbers. Consider the expression sqrt(p^3 * q) divided by sqrt[3]{p * q^4}. The...

1. TRANSLATE the radical expressions to exponential form

• Given expression: \(\sqrt{\mathrm{p}^3 \cdot \mathrm{q}} \div \sqrt[3]{\mathrm{p} \cdot \mathrm{q}^4}\)

• Convert to rational exponents:
  • \(\sqrt{\mathrm{p}^3 \cdot \mathrm{q}} = (\mathrm{p}^3 \cdot \mathrm{q})^{1/2} = \mathrm{p}^{3/2} \cdot \mathrm{q}^{1/2}\)
  • \(\sqrt[3]{\mathrm{p} \cdot \mathrm{q}^4} = (\mathrm{p} \cdot \mathrm{q}^4)^{1/3} = \mathrm{p}^{1/3} \cdot \mathrm{q}^{4/3}\)

• What this gives us: An expression we can work with using exponent rules

2. SIMPLIFY the division using exponent rules

• Set up the division: \([\mathrm{p}^{3/2} \cdot \mathrm{q}^{1/2}] \div [\mathrm{p}^{1/3} \cdot \mathrm{q}^{4/3}]\)

• Apply the rule \(\mathrm{a}^\mathrm{m} \div \mathrm{a}^\mathrm{n} = \mathrm{a}^{(\mathrm{m}-\mathrm{n})}\) to each variable:
  • For p: \(3/2 - 1/3 = 9/6 - 2/6 = 7/6\)
  • For q: \(1/2 - 4/3 = 3/6 - 8/6 = -5/6\)

• Result: \(\mathrm{p}^{7/6} \cdot \mathrm{q}^{-5/6}\)

3. TRANSLATE back to radical form

• Convert negative exponent: \(\mathrm{p}^{7/6} \cdot \mathrm{q}^{-5/6} = \mathrm{p}^{7/6} / \mathrm{q}^{5/6}\)

• Convert to sixth root notation: \(\sqrt[6]{\mathrm{p}^7} / \sqrt[6]{\mathrm{q}^5} = \sqrt[6]{\mathrm{p}^7/\mathrm{q}^5}\)

Answer: B. \(\sqrt[6]{\mathrm{p}^7/\mathrm{q}^5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when working with fractional exponents, particularly when finding common denominators.

For example, when calculating \(3/2 - 1/3\), they might incorrectly get \(2/1 = 2\) instead of \(7/6\), or when calculating \(1/2 - 4/3\), they might get \(-3/1 = -3\) instead of \(-5/6\). These errors lead to completely different final expressions that don't match any of the given choices, causing confusion and guessing.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students correctly convert to exponential form but fail to convert the final answer back to the radical form that matches the answer choices.

They might correctly arrive at \(\mathrm{p}^{7/6} \cdot \mathrm{q}^{-5/6}\) but then not recognize this needs to be written as \(\sqrt[6]{\mathrm{p}^7/\mathrm{q}^5}\). This leads them to think their work is wrong and abandon their systematic approach, resulting in random guessing.

The Bottom Line:

This problem requires fluent movement between radical and exponential notation combined with careful fraction arithmetic - students who struggle with either skill will find themselves unable to navigate to the correct answer systematically.