If a and c are positive numbers, which of the following is equivalent to \(\sqrt{(\mathrm{a} + \mathrm{c})^3 \cdot \sqrt{\mathrm{a} +...

1. TRANSLATE the problem information

• Given expression: \(\sqrt{(a + c)^3 \cdot \sqrt{a + c}}\)
• Need to find equivalent form from the answer choices
• Given: a and c are positive numbers

2. INFER the approach needed

• Recognize this involves properties of radicals and exponents
• Strategy: Simplify the expression inside the square root first
• Key insight: \(\sqrt{a + c}\) can be written as \((a + c)\) when considering the overall structure

3. SIMPLIFY the expression inside the radical

• Inside the main square root: \((a + c)^3 \cdot \sqrt{a + c}\)
• Treat \(\sqrt{a + c}\) as \((a + c)\): \((a + c)^3 \cdot (a + c) = (a + c)^4\)
• So the expression becomes: \(\sqrt{(a + c)^4}\)

4. SIMPLIFY the square root

• \(\sqrt{(a + c)^4} = \sqrt{((a + c)^2)^2} = (a + c)^2\)
• Since a and c are positive, \((a + c)^2 \gt 0\)

5. SIMPLIFY by expanding

• \((a + c)^2 = a^2 + 2ac + c^2\)
• This matches answer choice C

Answer: C. \(a^2 + 2ac + c^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with the nested radical structure and make errors combining the exponents. They might incorrectly treat \(\sqrt{a + c}\) as \((a + c)^{1/2}\) throughout, leading to \((a + c)^3 \cdot (a + c)^{1/2} = (a + c)^{7/2}\). Taking the square root gives \((a + c)^{7/4}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to expand \((a + c)^2\), but they stop at that step without first simplifying the radical expression properly. They might jump straight to expanding \((a + c)^2\) without establishing that this is indeed what the original expression equals. This may lead them to select Choice C by coincidence rather than through proper reasoning.

The Bottom Line:

This problem requires careful attention to the order of operations within nested radical expressions and confidence with exponent rules. Students must resist the urge to treat every square root as a fractional exponent and instead recognize when expressions can be simplified more directly.