Question:Which expression is equivalent to x^(2/3), where x gt 0?

1. TRANSLATE the fractional exponent to radical form

• Given: \(\mathrm{x^{2/3}}\) where \(\mathrm{x \gt 0}\)
• Need to find equivalent radical expression

The key conversion rule is: \(\mathrm{x^{m/n} = (\sqrt[n]{x})^m}\)

• The denominator (n) becomes the index of the radical
• The numerator (m) becomes the exponent outside the radical

2. INFER how to apply the conversion rule

• For \(\mathrm{x^{2/3}}\):
  • Denominator is 3 → cube root (\(\mathrm{\sqrt[3]{x}}\))
  • Numerator is 2 → exponent of 2
• This gives us: \(\mathrm{x^{2/3} = (\sqrt[3]{x})^2}\)

3. SIMPLIFY by checking answer choices

• (A) \(\mathrm{\frac{2}{3} \cdot x}\) → This is just multiplication, not equivalent
• (B) \(\mathrm{(\sqrt[3]{x})^2}\) → This matches our conversion!
• (C) \(\mathrm{2 \cdot \sqrt[3]{x}}\) → This is 2 times the cube root, not equivalent
• (D) \(\mathrm{\sqrt[3]{2x}}\) → This is cube root of 2x, not equivalent

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Fractional exponent conversion rule
Students who don't know or remember the rule \(\mathrm{x^{m/n} = (\sqrt[n]{x})^m}\) may try to guess based on pattern recognition or incorrectly apply arithmetic rules they know. They might think \(\mathrm{x^{2/3}}\) means "2/3 times x" because they're familiar with rules like \(\mathrm{3x}\) meaning "3 times x."
This may lead them to select Choice A (\(\mathrm{\frac{2}{3} \cdot x}\))

Second Most Common Error:

Weak SIMPLIFY execution: Confusing numerator and denominator roles
Students who know the conversion rule exists but mix up which part becomes the index versus the exponent. They might think the numerator 2 becomes the radical index and denominator 3 becomes the exponent, giving them \(\mathrm{\sqrt{x^3}}\) or looking for something like \(\mathrm{\sqrt[3]{x^2}}\) instead of \(\mathrm{(\sqrt[3]{x})^2}\).
This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can fluently convert between exponential and radical notation. Success depends on knowing the specific conversion rule and applying it correctly - there's no way to reason through this without the foundational knowledge.