If {36}{sqrt[3]{9y}} = 12, what is the value of 5y?591527

1. TRANSLATE the problem information

• Given equation: \(\frac{36}{\sqrt[3]{9y}} = 12\)
• Need to find: The value of \(5y\)

2. INFER the solution strategy

• To solve for expressions involving y, I need to eliminate the cube root first
• Strategy: Isolate the cube root term, then cube both sides to eliminate it

3. SIMPLIFY to isolate the cube root

• Starting with: \(\frac{36}{\sqrt[3]{9y}} = 12\)
• Divide both sides by 12: \(\sqrt[3]{9y} = \frac{36}{12} = 3\)

4. SIMPLIFY to eliminate the cube root

• Cube both sides: \((\sqrt[3]{9y})^3 = 3^3\)
• This gives us: \(9y = 27\)

5. INFER an efficient approach to find 5y

• Rather than solving for y first, I can use proportional reasoning
• If \(9y = 27\), then \(5y = \frac{5}{9} \times 27\)

6. SIMPLIFY the final calculation

• \(5y = \frac{5}{9} \times 27 = 5 \times 3 = 15\)

Answer: C (15)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the multi-step algebraic process, particularly when calculating \(36 \div 12\) or when cubing 3.

For example, if they incorrectly calculate \(36 \div 12 = 4\), they get \(\sqrt[3]{9y} = 4\), leading to \(9y = 64\) and completely wrong final values. Arithmetic mistakes at any step compound through the remaining calculations.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students get to \(9y = 27\) correctly but fail to recognize they need to find \(5y\), not just y or \(9y\).

Some students see \(9y = 27\) and think this is their final answer, leading them to select Choice D (27). Others solve \(y = 3\) but then think the question asks for just y, causing confusion since 3 isn't among the choices.

The Bottom Line:

This problem tests whether students can systematically work through multi-step radical equations while keeping track of what expression they actually need to find. Success requires both careful algebraic execution and strategic thinking about the most efficient path to the answer.