Let x be a positive real number. Which expression is equivalent to x^(-3/4)?

1. TRANSLATE the negative exponent using exponent rules

• Given: \(\mathrm{x^{-3/4}}\)
• The negative exponent rule tells us: \(\mathrm{x^{-a} = \frac{1}{x^a}}\)
• So: \(\mathrm{x^{-3/4} = \frac{1}{x^{3/4}}}\)

2. TRANSLATE the fractional exponent to radical form

• We now have: \(\mathrm{\frac{1}{x^{3/4}}}\)
• The fractional exponent rule tells us: \(\mathrm{x^{m/n} = \sqrt[n]{x^m}}\)
• So: \(\mathrm{x^{3/4} = \sqrt[4]{x^3}}\)
• Therefore: \(\mathrm{x^{-3/4} = \frac{1}{\sqrt[4]{x^3}}}\)

3. SIMPLIFY by checking our result against the answer choices

• Our result: \(\mathrm{\frac{1}{\sqrt[4]{x^3}}}\)
• This matches choice (B) exactly

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students forget that negative exponents mean reciprocals and only convert the fractional part to radical form.

They see \(\mathrm{x^{-3/4}}\) and immediately think "3/4 means fourth root of x cubed" without handling the negative sign. This gives them \(\mathrm{\sqrt[4]{x^3}}\), which is actually \(\mathrm{x^{3/4}}\), not \(\mathrm{x^{-3/4}}\).

This leads them to select Choice A (\(\mathrm{\sqrt[4]{x^3}}\))

Second Most Common Error:

Conceptual confusion about radical notation: Students misapply the fractional exponent rule by using the wrong root index.

They correctly get \(\mathrm{x^{-3/4} = \frac{1}{x^{3/4}}}\), but then incorrectly think \(\mathrm{x^{3/4} = \sqrt{x^3}}\) (using square root instead of fourth root). This gives them \(\mathrm{\frac{1}{\sqrt{x^3}}}\).

This may lead them to select Choice C (\(\mathrm{\frac{1}{\sqrt{x^3}}}\))

The Bottom Line:

This problem requires careful step-by-step application of exponent rules. Students must first handle the negative exponent (reciprocal), then convert the fractional exponent to radical form. Missing either step or applying rules in wrong order leads to incorrect answers.