sqrt(y/3) = 6What is the value of y in the equation above?

1. TRANSLATE the problem information

• Given equation: \(\sqrt{\frac{\mathrm{y}}{3}} = 6\)
• Goal: Find the value of y

2. INFER the solving strategy

• To solve for y, we need to eliminate the square root
• The most effective approach is to square both sides of the equation
• This will remove the radical and give us a linear equation to solve

3. SIMPLIFY by squaring both sides

• Square the left side: \((\sqrt{\frac{\mathrm{y}}{3}})^2 = \frac{\mathrm{y}}{3}\)
• Square the right side: \(6^2 = 36\)
• New equation: \(\frac{\mathrm{y}}{3} = 36\)

4. SIMPLIFY to isolate y

• Multiply both sides by 3: \(\mathrm{y} = 36 \times 3\)
• Calculate: \(\mathrm{y} = 108\)

5. Verify the solution

• Substitute back: \(\sqrt{\frac{108}{3}} = \sqrt{36} = 6\) ✓

Answer: C (108)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve without first eliminating the square root, trying to work directly with the radical expression or incorrectly applying operations.

For example, they might try to multiply both sides by 3 first, getting \(\sqrt{\mathrm{y}} = 18\), then struggle with how to continue. Or they might incorrectly think \(\sqrt{\frac{\mathrm{y}}{3}} \times 3 = \sqrt{3\mathrm{y}}\), leading to computational confusion and potentially selecting Choice A (72) through faulty arithmetic.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to square both sides but make arithmetic errors in the process.

They might forget to square the 6 (leaving 6 instead of 36) or make multiplication errors when calculating \(36 \times 3\). These calculation mistakes can lead them to select Choice B (36) if they stop at \(\frac{\mathrm{y}}{3} = 36\), or Choice D (324) if they incorrectly calculate the final multiplication.

The Bottom Line:

This problem tests whether students can systematically eliminate radicals through inverse operations. The key insight is recognizing that squaring both sides is the first necessary step, followed by careful arithmetic execution.