If a neq 0, which of the following expressions is equivalent to \(\frac{(27\mathrm{a}^9\mathrm{b}^6)^{1/3}}{\mathrm{a}^5}\)?

1. TRANSLATE the problem information

• Given expression: \((27\mathrm{a}^9\mathrm{b}^6)^{1/3} / \mathrm{a}^5\)
• Need to: Simplify to match one of the answer choices

2. INFER the approach

• The fractional exponent \((1/3)\) means we're taking the cube root of everything inside
• We should break this apart using the rule that \((\mathrm{abc})^\mathrm{n} = \mathrm{a}^\mathrm{n} \times \mathrm{b}^\mathrm{n} \times \mathrm{c}^\mathrm{n}\)
• Then we'll need to divide by \(\mathrm{a}^5\) using exponent division rules

3. SIMPLIFY the fractional exponent

Break apart \((27\mathrm{a}^9\mathrm{b}^6)^{1/3}\):

• \((27\mathrm{a}^9\mathrm{b}^6)^{1/3} = 27^{1/3} \times (\mathrm{a}^9)^{1/3} \times (\mathrm{b}^6)^{1/3}\)

4. SIMPLIFY each component

• \(27^{1/3} = 3\) (since \(3 \times 3 \times 3 = 27\))
• \((\mathrm{a}^9)^{1/3} = \mathrm{a}^{9 \times 1/3} = \mathrm{a}^3\)
• \((\mathrm{b}^6)^{1/3} = \mathrm{b}^{6 \times 1/3} = \mathrm{b}^2\)

So we have: \(3\mathrm{a}^3\mathrm{b}^2\)

5. SIMPLIFY the division

Divide \(3\mathrm{a}^3\mathrm{b}^2\) by \(\mathrm{a}^5\):

• \(3\mathrm{a}^3\mathrm{b}^2 / \mathrm{a}^5\)
• \(= 3\mathrm{b}^2 \times \mathrm{a}^{3-5}\)
• \(= 3\mathrm{b}^2 \times \mathrm{a}^{-2}\)
• \(= 3\mathrm{a}^{-2}\mathrm{b}^2\)

Answer: A (\(3\mathrm{a}^{-2}\mathrm{b}^2\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly apply the fractional exponent rule, treating \((27\mathrm{a}^9\mathrm{b}^6)^{1/3}\) as just taking the cube root of 27, forgetting to apply it to the variables.

They might get: \(3\mathrm{a}^9\mathrm{b}^6 / \mathrm{a}^5 = 3\mathrm{a}^4\mathrm{b}^6\)

This may lead them to select Choice B (\(3\mathrm{a}^3\mathrm{b}^2\)) or get confused and guess.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly break apart the fractional exponent but make arithmetic errors when subtracting exponents in the division step.

They might calculate: \(\mathrm{a}^3 / \mathrm{a}^5 = \mathrm{a}^{3+5} = \mathrm{a}^8\) (adding instead of subtracting)

This leads to confusion since \(\mathrm{a}^8\) doesn't appear in any answer choice, causing them to get stuck and guess.

The Bottom Line:

This problem requires careful, systematic application of multiple exponent rules in sequence. The key insight is recognizing that fractional exponents distribute across all factors, and negative exponents are often the correct final form.