If 2sqrt[3]{3x} = 6, what is the value of 4x?

1. TRANSLATE the problem information

• Given equation: \(2\sqrt[3]{3\mathrm{x}} = 6\)
• Need to find: the value of \(4\mathrm{x}\)

2. INFER the solution strategy

• We need to find \(\mathrm{x}\) first, then multiply by 4
• Work systematically to isolate the cube root, then eliminate it

3. SIMPLIFY by isolating the cube root

• Divide both sides by 2:

\(2\sqrt[3]{3\mathrm{x}} = 6\)

\(\sqrt[3]{3\mathrm{x}} = 3\)

4. SIMPLIFY by eliminating the cube root

• Cube both sides (since cubing and cube root are inverse operations):

\((\sqrt[3]{3\mathrm{x}})^3 = 3^3\)

\(3\mathrm{x} = 27\)

5. SIMPLIFY to solve for x

• Divide both sides by 3:

\(\mathrm{x} = 9\)

6. SIMPLIFY to find the final answer

• Multiply by 4:

\(4\mathrm{x} = 4(9) = 36\)

Answer: C. 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to work directly with \(4\mathrm{x}\) instead of finding \(\mathrm{x}\) first. They might attempt to manipulate the original equation to get \(4\mathrm{x}\) on one side, leading to unnecessary complexity and confusion. This often causes them to get stuck and abandon systematic solution, leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make errors when cubing both sides, such as incorrectly thinking \((\sqrt[3]{3\mathrm{x}})^3 = \sqrt[3]{3\mathrm{x}^3}\) or forgetting that \(3^3 = 27\). This leads to incorrect intermediate steps and wrong final calculations. This may lead them to select Choice A (12) or Choice B (27) depending on where the error occurred.

The Bottom Line:

This problem tests whether students can work systematically through multi-step algebraic manipulation involving cube roots, requiring them to resist the temptation to shortcut the process and instead follow logical step-by-step isolation of the variable.