Which of the following is equivalent to sqrt[4]{x^2 + 8x + 16}, where x gt 0?

1. TRANSLATE the radical notation

• Given: \(\sqrt[4]{x^2 + 8x + 16}\)
• This can be rewritten as: \((x^2 + 8x + 16)^{1/4}\)

2. INFER that factoring is the key strategy

• The expression under the radical is a trinomial
• Look for patterns - this might be a perfect square trinomial
• Check if \(x^2 + 8x + 16\) fits the pattern \(a^2 + 2ab + b^2 = (a + b)^2\)

3. SIMPLIFY by factoring the trinomial

• \(x^2 + 8x + 16 = (x + 4)^2\)
• Verify: \((x + 4)^2 = x^2 + 8x + 16\) ✓
• Now our expression becomes: \(((x + 4)^2)^{1/4}\)

4. SIMPLIFY using exponent rules

• Apply the rule: \((a^m)^n = a^{mn}\)
• \(((x + 4)^2)^{1/4} = (x + 4)^{2 \times 1/4} = (x + 4)^{1/2}\)

Answer: D. \((x + 4)^{1/2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(x^2 + 8x + 16\) is a perfect square trinomial

Students may try to work directly with the fourth root without factoring first, or they might not see the trinomial pattern. They may attempt to take the fourth root of each term separately or get confused about how to proceed without the factorization step. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Incorrectly applying exponent rules when simplifying \(((x + 4)^2)^{1/4}\)

Students might make errors like:

• \(((x + 4)^2)^{1/4} = (x + 4)^{2 + 1/4} = (x + 4)^{9/4}\)
• \(((x + 4)^2)^{1/4} = (x + 4)^{2/4} = (x + 4)^{2}\)

These calculation errors may lead them to select Choice A \((x + 4)^4\) or Choice B \((x + 4)^2\).

The Bottom Line:

This problem requires students to see beyond the complex-looking radical expression and recognize the underlying algebraic structure. The key insight is that factoring first makes the exponent simplification straightforward.