Question:\(\sqrt{\mathrm{a}}(4\sqrt{\mathrm{a}} - 2) - (3\mathrm{a} + 5)\)Which of the following is equivalent to the expression above for a gt 0?a...

1. INFER the approach needed

• This expression has two main parts: \(\sqrt{\mathrm{a}}(4\sqrt{\mathrm{a}} - 2)\) and \(-(3\mathrm{a} + 5)\)
• Strategy: Distribute first, then combine like terms
• Key insight: We need to handle the radical multiplication carefully

2. SIMPLIFY the first part using distribution

• Distribute \(\sqrt{\mathrm{a}}\) to each term: \(\sqrt{\mathrm{a}}(4\sqrt{\mathrm{a}} - 2)\)
• \(\sqrt{\mathrm{a}} \cdot 4\sqrt{\mathrm{a}} = 4(\sqrt{\mathrm{a}} \cdot \sqrt{\mathrm{a}}) = 4\mathrm{a}\) (since \(\sqrt{\mathrm{a}} \cdot \sqrt{\mathrm{a}} = \mathrm{a}\) for \(\mathrm{a} \gt 0\))
• \(\sqrt{\mathrm{a}} \cdot (-2) = -2\sqrt{\mathrm{a}}\)
• Result: \(4\mathrm{a} - 2\sqrt{\mathrm{a}}\)

3. SIMPLIFY the second part

• Distribute the negative sign: \(-(3\mathrm{a} + 5)\)
• \(-(3\mathrm{a}) = -3\mathrm{a}\)
• \(-(5) = -5\)
• Result: \(-3\mathrm{a} - 5\)

4. SIMPLIFY by combining all terms

• Full expression: \((4\mathrm{a} - 2\sqrt{\mathrm{a}}) + (-3\mathrm{a} - 5)\)
• Rewrite: \(4\mathrm{a} - 2\sqrt{\mathrm{a}} - 3\mathrm{a} - 5\)
• Combine like terms: \((4\mathrm{a} - 3\mathrm{a}) - 2\sqrt{\mathrm{a}} - 5\)
• Final result: \(\mathrm{a} - 2\sqrt{\mathrm{a}} - 5\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative distribution: Students correctly handle the first part but make sign errors when distributing the negative sign to \((3\mathrm{a} + 5)\). They might write \(-(3\mathrm{a} + 5)\) as \(-3\mathrm{a} + 5\), keeping the 5 positive instead of making it \(-5\).

This leads to: \(\mathrm{a} - 2\sqrt{\mathrm{a}} + 5\) instead of \(\mathrm{a} - 2\sqrt{\mathrm{a}} - 5\)

This may lead them to select Choice C (\(\mathrm{a} - 2\sqrt{\mathrm{a}} + 5\))

Second Most Common Error:

Poor SIMPLIFY execution with like terms: Students correctly distribute both parts but fail to combine the a terms properly. They might keep \(4\mathrm{a} - 3\mathrm{a}\) as separate terms rather than combining them to get \(\mathrm{a}\).

This leads to: \(4\mathrm{a} - 2\sqrt{\mathrm{a}} - 5\) instead of \(\mathrm{a} - 2\sqrt{\mathrm{a}} - 5\)

This may lead them to select Choice D (\(4\mathrm{a} - 2\sqrt{\mathrm{a}} + 5\))

The Bottom Line:

This problem tests careful algebraic manipulation skills, particularly with distributing negatives and combining like terms. The presence of both radical and polynomial terms requires students to stay organized and methodical throughout the multi-step simplification process.