Value Assessment Challenge: Finding the Greatest Number

To solve this problem, we'll follow these steps:

• Step 1: Simplify each choice using the Square Root Quotient Property.
• Step 2: Calculate the square root of each simplified fraction.
• Step 3: Compare these values to find the largest one.

Now, let's work through each step:

Step 1: Simplify each choice:
Choice 1: $\sqrt{\frac{36}{4}} = \frac{\sqrt{36}}{\sqrt{4}} = \frac{6}{2} = 3$
Choice 2: $\sqrt{\frac{36}{16}} = \frac{\sqrt{36}}{\sqrt{16}} = \frac{6}{4} = 1.5$
Choice 3: $\sqrt{\frac{36}{6}} = \frac{\sqrt{36}}{\sqrt{6}} = \frac{6}{\sqrt{6}} \approx 2.45$ (approximating since it is irrational)
Choice 4: $\sqrt{\frac{36}{2}} = \frac{\sqrt{36}}{\sqrt{2}} = \frac{6}{\sqrt{2}} = \frac{6 \times \sqrt{2}}{2} = 3\sqrt{2} \approx 4.24$

Step 2: Evaluate to find which is the largest value:
Choice 1 equals 3, Choice 2 equals 1.5, Choice 3 equals approximately 2.45, and Choice 4 approximately equals 4.24.

Step 3: Compare these values. Clearly, $4.24$ is the largest.

Therefore, the solution to the problem is $\sqrt{\frac{36}{2}}$.