Question:Which expression is equivalent to x^(5/6) * x^(1/3), where x gt 0?sqrt[3]{x^7}sqrt[6]{x^7}sqrt[7]{x^6}sqrt[6]{x^6}

1. INFER the strategy needed

• We have multiplication of powers with the same base: \(\mathrm{x^{5/6} \cdot x^{1/3}}\)
• This calls for the product rule for exponents
• The answer choices are all in radical form, so we'll need to convert our final result

2. SIMPLIFY by applying the product rule

• Use \(\mathrm{x^a \cdot x^b = x^{(a+b)}}\)
• \(\mathrm{x^{5/6} \cdot x^{1/3} = x^{(5/6 + 1/3)}}\)
• Need to add the exponents: \(\mathrm{\frac{5}{6} + \frac{1}{3}}\)

3. SIMPLIFY the fraction addition

• Find common denominator for \(\mathrm{\frac{5}{6} + \frac{1}{3}}\)
• Convert: \(\mathrm{\frac{1}{3} = \frac{2}{6}}\)
• Add: \(\mathrm{\frac{5}{6} + \frac{2}{6} = \frac{7}{6}}\)
• Result: \(\mathrm{x^{7/6}}\)

4. SIMPLIFY by converting to radical form

• Use \(\mathrm{x^{m/n} = \sqrt[n]{x^m}}\)
• \(\mathrm{x^{7/6} = \sqrt[6]{x^7}}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors when adding fractions with different denominators. They might incorrectly add \(\mathrm{\frac{5}{6} + \frac{1}{3}}\) as \(\mathrm{\frac{6}{9}}\) or \(\mathrm{\frac{5}{9}}\), thinking they can just add numerators and denominators separately.

This leads to wrong exponents like \(\mathrm{x^{6/9} = x^{2/3}}\) or \(\mathrm{x^{5/9}}\), which when converted to radical form don't match any answer choice. This causes confusion and may lead them to select Choice A (\(\mathrm{\sqrt[3]{x^7}}\)) thinking it "looks closest" to what they expect.

Second Most Common Error:

Conceptual confusion about radical conversion: Students correctly get \(\mathrm{x^{7/6}}\) but then misapply the radical conversion rule. They might think \(\mathrm{x^{7/6} = \sqrt[7]{x^6}}\) instead of \(\mathrm{\sqrt[6]{x^7}}\), mixing up which number goes where.

This may lead them to select Choice C (\(\mathrm{\sqrt[7]{x^6}}\)).

The Bottom Line:

This problem tests whether students can smoothly combine fraction arithmetic with exponent rules. The key challenge is maintaining accuracy through multiple algebraic steps while keeping track of which part of a rational exponent becomes which part of a radical.