Which expression is equivalent to x^(3/4), for x gt 0?sqrt[6]{x^(18)}sqrt[4]{x^(12)}sqrt[12]{x^(16)}sqrt[32]{x^(24)}

1. TRANSLATE the problem information

• Given: Need to find which expression equals \(\mathrm{x}^{3/4}\)
• Answer choices: Four different radical expressions
• What this tells us: We need to convert radicals to exponential form for comparison

2. TRANSLATE each radical using the conversion rule

• Use the rule: \(\sqrt[n]{\mathrm{a}^b} = \mathrm{a}^{b/n}\)
• This means the exponent inside becomes the numerator, and the index becomes the denominator

3. SIMPLIFY each converted expression

• Choice A: \(\sqrt[6]{\mathrm{x}^{18}} = \mathrm{x}^{18/6} = \mathrm{x}^3\)
• Choice B: \(\sqrt[4]{\mathrm{x}^{12}} = \mathrm{x}^{12/4} = \mathrm{x}^3\)
• Choice C: \(\sqrt[12]{\mathrm{x}^{16}} = \mathrm{x}^{16/12} = \mathrm{x}^{4/3}\)
• Choice D: \(\sqrt[32]{\mathrm{x}^{24}} = \mathrm{x}^{24/32} = \mathrm{x}^{3/4}\)

4. APPLY CONSTRAINTS to select the correct answer

• We need the expression that equals \(\mathrm{x}^{3/4}\)
• Only choice D gives us \(\mathrm{x}^{3/4}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the conversion rule and incorrectly think \(\sqrt[n]{\mathrm{a}^b} = \mathrm{a}^{n/b}\) instead of \(\mathrm{a}^{b/n}\).

Using the wrong rule, they might calculate:

• Choice A: \(\sqrt[6]{\mathrm{x}^{18}} = \mathrm{x}^{6/18} = \mathrm{x}^{1/3}\)
• Choice D: \(\sqrt[32]{\mathrm{x}^{24}} = \mathrm{x}^{32/24} = \mathrm{x}^{4/3}\)

This leads to confusion since none of their calculations match \(\mathrm{x}^{3/4}\), causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly apply the conversion rule but make arithmetic errors when reducing fractions.

For example, they might incorrectly simplify \(24/32\) as \(6/8\) instead of \(3/4\), or miscalculate \(16/12\) as something other than \(4/3\).

This may lead them to select Choice A or B (both \(\mathrm{x}^3\)) thinking one of these massive simplification errors somehow equals 3/4.

The Bottom Line:

This problem tests whether students can correctly apply the radical-exponential conversion rule and perform accurate fraction arithmetic - both fundamental skills for working with rational exponents.