The expression {sqrt[3]{m^(12) n^6}}{m^2 sqrt(n^8)} is equivalent to m^a n^b for some constants a and b. If m gt 1...

1. TRANSLATE the radical expressions to fractional exponents

• Given expression: \(\frac{\sqrt[3]{\mathrm{m}^{12} \mathrm{n}^6}}{\mathrm{m}^2 \sqrt{\mathrm{n}^8}}\)
• Key conversion rules:
  • Cube root: \(\sqrt[3]{\mathrm{x}} = \mathrm{x}^{1/3}\)
  • Square root: \(\sqrt{\mathrm{x}} = \mathrm{x}^{1/2}\)

2. SIMPLIFY the numerator

• \(\sqrt[3]{\mathrm{m}^{12} \mathrm{n}^6} = (\mathrm{m}^{12} \mathrm{n}^6)^{1/3}\)
• Apply the fractional exponent to each factor:
  • \(\mathrm{m}^{12 \cdot 1/3} = \mathrm{m}^4\)
  • \(\mathrm{n}^{6 \cdot 1/3} = \mathrm{n}^2\)
• Result: \(\mathrm{m}^4 \mathrm{n}^2\)

3. SIMPLIFY the denominator

• \(\mathrm{m}^2 \sqrt{\mathrm{n}^8} = \mathrm{m}^2 (\mathrm{n}^8)^{1/2}\)
• Apply the fractional exponent:
  • \(\mathrm{n}^{8 \cdot 1/2} = \mathrm{n}^4\)
• Result: \(\mathrm{m}^2 \mathrm{n}^4\)

4. SIMPLIFY the overall fraction

• The expression becomes: \(\frac{\mathrm{m}^4 \mathrm{n}^2}{\mathrm{m}^2 \mathrm{n}^4}\)
• Apply quotient rule \(\frac{\mathrm{x}^\mathrm{a}}{\mathrm{x}^\mathrm{b}} = \mathrm{x}^{\mathrm{a}-\mathrm{b}}\):
  • For m: \(\frac{\mathrm{m}^4}{\mathrm{m}^2} = \mathrm{m}^{4-2} = \mathrm{m}^2\)
  • For n: \(\frac{\mathrm{n}^2}{\mathrm{n}^4} = \mathrm{n}^{2-4} = \mathrm{n}^{-2}\)

5. INFER the values of a and b

• The simplified expression is \(\mathrm{m}^2 \mathrm{n}^{-2}\)
• Comparing to \(\mathrm{m}^\mathrm{a} \mathrm{n}^\mathrm{b}\): \(\mathrm{a} = 2\) and \(\mathrm{b} = -2\)
• Therefore: \(\mathrm{a} - \mathrm{b} = 2 - (-2) = 4\)

Answer: D (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when working with negative exponents or incorrectly apply the quotient rule. For example, they might calculate \(\frac{\mathrm{n}^2}{\mathrm{n}^4} = \mathrm{n}^{2+4} = \mathrm{n}^6\) instead of \(\mathrm{n}^{2-4} = \mathrm{n}^{-2}\), leading to \(\mathrm{b} = 6\) instead of \(\mathrm{b} = -2\).

This error would give \(\mathrm{a} - \mathrm{b} = 2 - 6 = -4\), leading them to select Choice A (-4).

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students struggle to correctly convert the radical expressions to fractional exponents, particularly with the cube root. They might incorrectly write \(\sqrt[3]{\mathrm{m}^{12}} = \mathrm{m}^{12/2} = \mathrm{m}^6\) instead of \(\mathrm{m}^{12/3} = \mathrm{m}^4\).

This leads to incorrect values throughout the problem, causing confusion and potentially guessing among the remaining choices.

The Bottom Line:

This problem requires careful attention to exponent arithmetic and sign handling. The negative exponent in the final answer is the key trap - students who rush through the quotient rule often miss the subtraction that creates the negative exponent.