Which expression is equivalent to x^(1/4), where x gt 0?

1. INFER the mathematical relationship needed

• Given: \(\mathrm{x^{1/4}}\) where \(\mathrm{x \gt 0}\)
• Need to find equivalent expression
• This requires applying the fractional exponent rule: \(\mathrm{x^{1/n} = \sqrt[n]{x}}\)

2. INFER by applying the rule directly

• Using \(\mathrm{x^{1/n} = \sqrt[n]{x}}\) with \(\mathrm{n = 4}\):
• \(\mathrm{x^{1/4} = \sqrt[4]{x}}\)

3. INFER by checking answer choices

• Choice A: \(\mathrm{\frac{1}{14} \cdot x = \frac{x}{14}}\) (this is not equivalent)
• Choice B: \(\mathrm{\sqrt[4]{x}}\) (this matches our result!)
• Choice C: \(\mathrm{14 \cdot x = 14x}\) (this is not equivalent)
• Choice D: \(\mathrm{\frac{1}{x}}\) (this is not equivalent)

Answer: B. \(\mathrm{\sqrt[4]{x}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't remember or understand the fractional exponent rule \(\mathrm{x^{1/n} = \sqrt[n]{x}}\)

Without this fundamental rule, students may try to interpret \(\mathrm{x^{1/4}}\) using incorrect reasoning like:

• Thinking 1/4 means "divide by 4" leading to \(\mathrm{\frac{x}{4}}\) (close to Choice A)
• Thinking the exponent 1/4 creates a reciprocal, leading to \(\mathrm{\frac{1}{x}}\) (Choice D)
• Complete confusion about what fractional exponents mean

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests direct recall and application of a fundamental exponent rule. Success depends entirely on knowing that fractional exponents convert to radicals using \(\mathrm{x^{1/n} = \sqrt[n]{x}}\).