Let p and q be real numbers with p neq 0. Which of the following expressions is equivalent to sqrt[3]{64p^9q^6/8p^3}?

1. SIMPLIFY the expression inside the cube root first

• Given: \(\sqrt[3]{\frac{64\mathrm{p}^9\mathrm{q}^6}{8\mathrm{p}^3}}\)

• Before taking any cube root, simplify the fraction:
  • Coefficients: \(64/8 = 8\)
  • p terms: \(\mathrm{p}^9/\mathrm{p}^3 = \mathrm{p}^{(9-3)} = \mathrm{p}^6\)
  • q terms: \(\mathrm{q}^6/1 = \mathrm{q}^6\)

• This gives us: \(\sqrt[3]{8\mathrm{p}^6\mathrm{q}^6}\)

2. INFER that cube roots can be separated across multiplication

• Key insight: \(\sqrt[3]{\mathrm{abc}} = \sqrt[3]{\mathrm{a}} \times \sqrt[3]{\mathrm{b}} \times \sqrt[3]{\mathrm{c}}\)

• Apply this property: \(\sqrt[3]{8\mathrm{p}^6\mathrm{q}^6} = \sqrt[3]{8} \times \sqrt[3]{\mathrm{p}^6} \times \sqrt[3]{\mathrm{q}^6}\)

3. SIMPLIFY each cube root separately

• \(\sqrt[3]{8} = 2\) (since \(2^3 = 8\))
• \(\sqrt[3]{\mathrm{p}^6} = \mathrm{p}^{6/3} = \mathrm{p}^2\)
• \(\sqrt[3]{\mathrm{q}^6} = \mathrm{q}^{6/3} = \mathrm{q}^2\)

• Multiply the results: \(2 \times \mathrm{p}^2 \times \mathrm{q}^2 = 2\mathrm{p}^2\mathrm{q}^2\)

Answer: B. \(2\mathrm{p}^2\mathrm{q}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students attempt to take the cube root without first simplifying the fraction inside the radical.

They might try to work with \(\sqrt[3]{\frac{64\mathrm{p}^9\mathrm{q}^6}{8\mathrm{p}^3}}\) directly, leading to confusion about how to handle the division within a cube root. This complicated approach often results in calculation errors or abandoning systematic solution.

This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about exponent rules: Students correctly simplify to \(\sqrt[3]{8\mathrm{p}^6\mathrm{q}^6}\) but make errors when applying cube roots to the variable terms.

Common mistakes include \(\sqrt[3]{\mathrm{p}^6} = \mathrm{p}^3\) (forgetting to divide the exponent by 3) or \(\sqrt[3]{\mathrm{q}^6} = \mathrm{q}\) (similar error). These exponent errors can lead them to select Choice D (\(4\mathrm{p}^3\mathrm{q}^2\)) if they also miscalculate \(\sqrt[3]{8}\) as 4, or other incorrect combinations.

The Bottom Line:

This problem tests whether students can systematically apply algebraic simplification before attempting radical operations. The key is recognizing that complex expressions under radicals should be simplified first using basic exponent rules.