Which expression is equivalent to a^(11/12), where a gt 0?

1. TRANSLATE the problem information

• Given: \(\mathrm{a}^{11/12}\) where \(\mathrm{a} \gt 0\)
• Need to find: Which radical expression is equivalent
• Answer choices are all in the form \(^n\sqrt{\mathrm{a}^{132}}\)

2. INFER the solution strategy

• Since we're comparing expressions with different forms (exponential vs radical), convert everything to the same form
• Convert all radical expressions to exponential form using \(^n\sqrt{\mathrm{m}} = \mathrm{m}^{1/n}\)
• Then compare exponents directly

3. SIMPLIFY each answer choice to exponential form

Convert each radical using \(^n\sqrt{\mathrm{m}} = \mathrm{m}^{1/n}\):

Choice A: \(^3\sqrt{\mathrm{a}^{132}}\) = \((\mathrm{a}^{132})^{1/3}\) = \(\mathrm{a}^{132/3}\) = \(\mathrm{a}^{44}\)

Choice B: \(^{144}\sqrt{\mathrm{a}^{132}}\) = \((\mathrm{a}^{132})^{1/144}\) = \(\mathrm{a}^{132/144}\)

Choice C: \(^{12}\sqrt{\mathrm{a}^{132}}\) = \((\mathrm{a}^{132})^{1/12}\) = \(\mathrm{a}^{132/12}\) = \(\mathrm{a}^{11}\)

Choice D: \(^{11}\sqrt{\mathrm{a}^{132}}\) = \((\mathrm{a}^{132})^{1/11}\) = \(\mathrm{a}^{132/11}\) = \(\mathrm{a}^{12}\)

4. SIMPLIFY the fraction in Choice B

For Choice B, we need to simplify \(132/144\):

• Find common factors: \(132 = 12 \times 11\) and \(144 = 12 \times 12\)
• Divide both by 12: \(132/144 = 11/12\)

Therefore: \(\mathrm{a}^{132/144} = \mathrm{a}^{11/12}\)

5. INFER the final answer

Choice B gives us \(\mathrm{a}^{11/12}\), which exactly matches our original expression.

Answer: B. \(^{144}\sqrt{\mathrm{a}^{132}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students convert the radicals correctly but make calculation errors when simplifying fractions like \(132/144\). They might not recognize that both numbers are divisible by 12, or make arithmetic mistakes in the division process. This leads to incorrect equivalence determinations and could cause them to select Choice A, C, or D instead of recognizing that only Choice B simplifies to match the original expression.

Second Most Common Error:

Missing conceptual knowledge about radical-exponent conversion: Students don't remember or correctly apply the relationship \(^n\sqrt{\mathrm{m}} = \mathrm{m}^{1/n}\). They might try to work with the radicals directly or use incorrect conversion rules, leading to confusion and inability to make progress. This causes them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem tests both conceptual knowledge of the radical-exponent relationship and the execution skills to work through multi-step algebraic simplification accurately. Success requires systematic conversion and careful fraction arithmetic.