The expression {sqrt(x) * sqrt[3]{x}}{sqrt[6]{x}} is equivalent to x^k for all positive values of x.What is the value of k?

1. TRANSLATE the radical expressions to exponential form

• Given information:
  • Expression: \(\frac{\sqrt{\mathrm{x}} \cdot \sqrt[3]{\mathrm{x}}}{\sqrt[6]{\mathrm{x}}}\)
  • Need to find: value of k where this equals \(\mathrm{x}^\mathrm{k}\)

• Using the conversion rule \(\sqrt[\mathrm{n}]{\mathrm{a}} = \mathrm{a}^{(1/\mathrm{n})}\):
  • \(\sqrt{\mathrm{x}} = \mathrm{x}^{(1/2)}\)
  • \(\sqrt[3]{\mathrm{x}} = \mathrm{x}^{(1/3)}\)
  • \(\sqrt[6]{\mathrm{x}} = \mathrm{x}^{(1/6)}\)

2. SIMPLIFY by substituting the exponential forms

• The expression becomes: \(\frac{\mathrm{x}^{(1/2)} \cdot \mathrm{x}^{(1/3)}}{\mathrm{x}^{(1/6)}}\)

3. SIMPLIFY the numerator using the product rule

• When multiplying powers with the same base, add the exponents:
  \(\mathrm{x}^{(1/2)} \cdot \mathrm{x}^{(1/3)} = \mathrm{x}^{(1/2 + 1/3)}\)

• To add these fractions, find common denominator 6:
  • \(\frac{1}{2} = \frac{3}{6}\)
  • \(\frac{1}{3} = \frac{2}{6}\)
  • So: \(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)

• Numerator becomes: \(\mathrm{x}^{(5/6)}\)

4. SIMPLIFY the entire expression using the quotient rule

• When dividing powers with the same base, subtract the exponents:
  \(\frac{\mathrm{x}^{(5/6)}}{\mathrm{x}^{(1/6)}} = \mathrm{x}^{(5/6 - 1/6)} = \mathrm{x}^{(4/6)}\)

5. SIMPLIFY the final exponent

• Reduce the fraction: \(\frac{4}{6} = \frac{2}{3}\)
• Therefore: \(\mathrm{k} = \frac{2}{3}\)

Answer: C) 2/3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly convert to exponential form but make arithmetic errors when adding fractions \(\frac{1}{2} + \frac{1}{3}\), often getting \(\frac{2}{5}\) instead of \(\frac{5}{6}\).

When they incorrectly get \(\frac{1}{2} + \frac{1}{3} = \frac{2}{5}\), their numerator becomes \(\mathrm{x}^{(2/5)}\), leading to:
\(\frac{\mathrm{x}^{(2/5)}}{\mathrm{x}^{(1/6)}} = \mathrm{x}^{(2/5 - 1/6)}\)

To subtract these fractions: \(\frac{2}{5} - \frac{1}{6} = \frac{12}{30} - \frac{5}{30} = \frac{7}{30}\)

This may lead them to select an answer not among the choices, causing confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students attempt to work directly with radical notation without converting to exponential form first, getting overwhelmed by the complex radical manipulations.

This leads them to abandon systematic solution and guess among the given choices.

The Bottom Line:

This problem tests whether students can systematically convert between radical and exponential notation, then apply exponent rules accurately. The key challenge is maintaining precision in fraction arithmetic while managing multiple algebraic steps.