If a neq 0, which of the following expressions is equivalent to \(\frac{(64\mathrm{a}^9\mathrm{b}^6)^{1/3}}{2\mathrm{a}^4\mathrm{b}}\)?

1. INFER the solution strategy

• We have a fraction with a cube root in the numerator and a polynomial in the denominator
• Strategy: First simplify the cube root, then divide the resulting expression

2. SIMPLIFY the cube root in the numerator

• Apply the cube root to each factor separately:
  \((64\mathrm{a}^9\mathrm{b}^6)^{1/3} = (64)^{1/3} \cdot (\mathrm{a}^9)^{1/3} \cdot (\mathrm{b}^6)^{1/3}\)

• Calculate each part:
  • \((64)^{1/3} = 4\) (since \(4^3 = 64\))
  • \((\mathrm{a}^9)^{1/3} = \mathrm{a}^{9/3} = \mathrm{a}^3\)
  • \((\mathrm{b}^6)^{1/3} = \mathrm{b}^{6/3} = \mathrm{b}^2\)

• The numerator becomes: \(4\mathrm{a}^3\mathrm{b}^2\)

3. SIMPLIFY the division

• Now we have: \(\frac{4\mathrm{a}^3\mathrm{b}^2}{2\mathrm{a}^4\mathrm{b}}\)

• Separate the coefficients and variables:
  \(= \frac{4}{2} \cdot \frac{\mathrm{a}^3}{\mathrm{a}^4} \cdot \frac{\mathrm{b}^2}{\mathrm{b}}\)

• Apply exponent rules:
  \(= 2 \cdot \mathrm{a}^{3-4} \cdot \mathrm{b}^{2-1}\)
  \(= 2 \cdot \mathrm{a}^{-1} \cdot \mathrm{b}^1\)
  \(= 2 \cdot \frac{1}{\mathrm{a}} \cdot \mathrm{b}\)
  \(= \frac{2\mathrm{b}}{\mathrm{a}}\)

Answer: C. \(\frac{2\mathrm{b}}{\mathrm{a}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors when applying the cube root to the entire expression rather than each factor separately, or they struggle with negative exponents.

For example, they might incorrectly calculate \((\mathrm{a}^9)^{1/3} = \mathrm{a}^2\) instead of \(\mathrm{a}^3\), or when they get \(\mathrm{a}^{-1}\), they might leave it as \(\mathrm{a}^{-1}\) instead of converting to \(\frac{1}{\mathrm{a}}\). This leads to incorrect final expressions that don't match any answer choice, causing them to guess randomly.

Second Most Common Error:

Insufficient INFER reasoning: Students recognize they need to work with the cube root but don't develop a clear strategy for handling the division step systematically.

They might correctly simplify the numerator to \(4\mathrm{a}^3\mathrm{b}^2\) but then struggle with organizing the division step, leading to calculation errors like forgetting to subtract exponents properly. This may lead them to select Choice A (\(\frac{\mathrm{b}}{4\mathrm{a}}\)) by incorrectly handling the coefficient division.

The Bottom Line:

This problem requires systematic application of multiple exponent rules in sequence. Students who try to do too many steps at once or who aren't comfortable with negative exponents will struggle to reach the correct simplified form.