If x gt -3, which of the following is equivalent to the expression \(\frac{\sqrt{\mathrm{x}^2 + 6\mathrm{x} + 9}}{(\mathrm{x} + 3)^{1/4}}\)?

1. TRANSLATE the problem information

• Given expression: \(\frac{\sqrt{\mathrm{x}^2 + 6\mathrm{x} + 9}}{(\mathrm{x} + 3)^{1/4}}\)
• Given constraint: \(\mathrm{x} \gt -3\)
• Need to find equivalent form from answer choices

2. INFER what to do with the numerator

• Look at \(\mathrm{x}^2 + 6\mathrm{x} + 9\) - this has the pattern of a perfect square trinomial
• Check: \((\mathrm{x}+3)^2 = \mathrm{x}^2 + 2(\mathrm{x})(3) + 3^2 = \mathrm{x}^2 + 6\mathrm{x} + 9\) ✓
• So \(\sqrt{\mathrm{x}^2 + 6\mathrm{x} + 9} = \sqrt{(\mathrm{x}+3)^2}\)

3. SIMPLIFY the square root using absolute value

• \(\sqrt{(\mathrm{x}+3)^2} = |\mathrm{x}+3|\)
• This is where the absolute value comes in - we need to determine the sign

4. APPLY CONSTRAINTS to resolve the absolute value

• Since \(\mathrm{x} \gt -3\), we have \(\mathrm{x} + 3 \gt 0\)
• When the expression inside absolute value bars is positive: \(|\mathrm{x}+3| = \mathrm{x}+3\)
• Our expression becomes: \(\frac{\mathrm{x}+3}{(\mathrm{x}+3)^{1/4}}\)

5. SIMPLIFY using exponent rules

• Rewrite as: \((\mathrm{x}+3)^1 \cdot (\mathrm{x}+3)^{-1/4}\)
• Apply the rule \(\mathrm{a}^\mathrm{m} \cdot \mathrm{a}^\mathrm{n} = \mathrm{a}^{\mathrm{m}+\mathrm{n}}\): \((\mathrm{x}+3)^{1-1/4} = (\mathrm{x}+3)^{3/4}\)

Answer: (C) \((\mathrm{x}+3)^{3/4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(\mathrm{x}^2 + 6\mathrm{x} + 9\) is a perfect square trinomial

Students might try to factor this as two separate terms or attempt to use the quadratic formula unnecessarily. Without seeing the pattern \((\mathrm{x}+3)^2\), they get stuck trying to simplify \(\sqrt{\mathrm{x}^2 + 6\mathrm{x} + 9}\) and may resort to leaving it as is or making calculation errors. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Ignoring the condition \(\mathrm{x} \gt -3\) when dealing with \(|\mathrm{x}+3|\)

Students correctly get to \(\sqrt{(\mathrm{x}+3)^2} = |\mathrm{x}+3|\) but then don't use the given constraint. They might leave the answer with absolute value bars or incorrectly assume \(|\mathrm{x}+3| = -(\mathrm{x}+3)\). This prevents them from simplifying to \((\mathrm{x}+3)\) and getting the correct final exponent. This may lead them to select Choice (A) \((\mathrm{x}+3)^{1/4}\) if they make further errors with exponent rules.

The Bottom Line:

This problem tests whether students can spot perfect square patterns and properly handle absolute values with constraints - two skills that require both pattern recognition and careful attention to given conditions.