If 4sqrt(2x) = 16, what is the value of 6x?

1. INFER the solution strategy

• Given: \(4\sqrt{2\mathrm{x}} = 16\)
• Goal: Find the value of \(6\mathrm{x}\)
• Strategy: Isolate the radical term first, then eliminate the square root

2. SIMPLIFY by isolating the radical

• Divide both sides by 4:

\(4\sqrt{2\mathrm{x}} \div 4 = 16 \div 4\)
\(\sqrt{2\mathrm{x}} = 4\)

3. SIMPLIFY by eliminating the square root

• Square both sides:

\((\sqrt{2\mathrm{x}})^2 = 4^2\)
\(2\mathrm{x} = 16\)

4. INFER how to find 6x from 2x

• We have \(2\mathrm{x} = 16\) and need \(6\mathrm{x}\)
• Since \(6\mathrm{x} = 3 \times (2\mathrm{x})\), multiply both sides by 3:

\(3(2\mathrm{x}) = 3(16)\)
\(6\mathrm{x} = 48\)

Answer: B. 48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the correct sequence of operations, leading to premature squaring of the entire equation \(4\sqrt{2\mathrm{x}} = 16\).

Students might square both sides immediately: \((4\sqrt{2\mathrm{x}})^2 = 16^2\), giving \(16(2\mathrm{x}) = 256\), then \(32\mathrm{x} = 256\), so \(\mathrm{x} = 8\). Then calculating \(6\mathrm{x} = 48\). While this approach coincidentally gives the right answer, it demonstrates poor understanding of radical equation solving and could lead to errors in more complex problems.

Second Most Common Error:

Poor SIMPLIFY execution: Solving correctly to find \(\mathrm{x} = 8\), but then calculating the wrong multiple of x.

Students correctly get \(2\mathrm{x} = 16\), so \(\mathrm{x} = 8\). But then they calculate \(3\mathrm{x} = 3(8) = 24\) instead of \(6\mathrm{x} = 6(8) = 48\). This leads them to select Choice A (24).

The Bottom Line:

This problem tests whether students can systematically work through a multi-step algebraic process while keeping track of what the question is actually asking for. The key insight is maintaining focus on the goal (finding \(6\mathrm{x}\)) throughout the solution process.