If 3sqrt[3]{4x} = 12, what is the value of 5x?20406080

1. TRANSLATE the problem information

• Given equation: \(3\sqrt[3]{4\mathrm{x}} = 12\)
• Need to find: The value of \(5\mathrm{x}\) (not just x)

2. INFER the approach

• To solve for x, we need to isolate it by first getting rid of the cube root
• Strategy: Isolate the cube root term, then eliminate it by cubing both sides

3. Isolate the cube root term

• Divide both sides by 3: \(\sqrt[3]{4\mathrm{x}} = 12 \div 3 = 4\)

4. SIMPLIFY by eliminating the cube root

• Cube both sides: \((\sqrt[3]{4\mathrm{x}})^3 = 4^3\)
• This gives us: \(4\mathrm{x} = 64\)
• (Remember: cubing and cube root are inverse operations, so they cancel out)

5. Solve for x

• Divide both sides by 4: \(\mathrm{x} = 64 \div 4 = 16\)

6. TRANSLATE the final requirement

• The question asks for \(5\mathrm{x}\), not just x
• Calculate: \(5\mathrm{x} = 5(16) = 80\)

Answer: D (80)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors, particularly when cubing 4, incorrectly calculating \(4^3\) as 16 instead of 64.

This leads them to get \(4\mathrm{x} = 16\), so \(\mathrm{x} = 4\), and therefore \(5\mathrm{x} = 20\). They may select Choice A (20).

Second Most Common Error:

Poor TRANSLATE reasoning: Students solve correctly through finding \(\mathrm{x} = 16\) but forget that the question asks for \(5\mathrm{x}\), not x.

They stop at \(\mathrm{x} = 16\) and look for this value among the choices. Since 16 isn't listed, they might make a guess or select the closest value, potentially choosing Choice A (20).

The Bottom Line:

This problem tests whether students can systematically work through inverse operations while maintaining accuracy in arithmetic and careful attention to what the question is actually asking for.