x^4 = 81Which of the following values of x satisfies the given equation?-{9}3927

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x^4 = 81}\)
• Need to find: Which value of x from the choices satisfies this equation

2. INFER the solution approach

• To solve \(\mathrm{x^4 = 81}\), I need to "undo" the fourth power
• The inverse operation of raising to the 4th power is taking the fourth root
• Strategy: Take the fourth root of both sides

3. SIMPLIFY by taking fourth roots

• \(\mathrm{x = \pm}\sqrt[4]{81}\)
• Need to evaluate \(\sqrt[4]{81}\): What number raised to the 4th power gives 81?
• Since \(\mathrm{3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81}\), we have \(\sqrt[4]{81} = 3\)
• Therefore: \(\mathrm{x = \pm 3}\) (both +3 and -3 are mathematically valid solutions)

4. APPLY CONSTRAINTS to select from answer choices

• The complete solution set is \(\{3, -3\}\)
• Checking answer choices: (A) -9, (B) 3, (C) 9, (D) 27
• Only \(\mathrm{x = 3}\) appears among the given options

5. Verify the answer

• Check: Does \(\mathrm{3^4 = 81}\)?
• \(\mathrm{3^4 = 3 \times 3 \times 3 \times 3 = 81}\) ✓

Answer: B (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to take the fourth root to solve \(\mathrm{x^4 = 81}\). Instead, they might try to guess-and-check with the answer choices without a systematic approach, or attempt inappropriate methods like factoring. This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt the correct strategy but make calculation errors when evaluating fourth powers. They might incorrectly calculate that \(\mathrm{9^4 = 81}\) (when actually \(\mathrm{9^4 = 6,561}\)) or make arithmetic mistakes when checking their work. This may lead them to select Choice C (9) based on faulty calculations.

The Bottom Line:

This problem tests whether students can systematically solve higher-order polynomial equations by applying inverse operations, rather than relying on guess-and-check methods. The key insight is recognizing that fourth roots are needed and being able to evaluate them accurately.