Finding the Largest Value: Numerical Comparison Exercise

To solve this problem, we'll follow the steps below:

• Simplify each mathematical expression using exponent rules.
• Compare the values derived from each simplification.

Let us analyze each given choice:

Choice 1: $\sqrt{\sqrt[6]{4}}$

• The expression is simplified as follows: $(\sqrt[6]{4})^{\frac{1}{2}} = 4^{\frac{1}{6} \times \frac{1}{2}} = 4^{\frac{1}{12}}$.

Choice 2: $\sqrt[6]{4}$

• This expression is: $4^{\frac{1}{6}}$.

Choice 3: $\sqrt[2]{\sqrt[3]{4}}$

• Simplified, this is: $(\sqrt[3]{4})^{\frac{1}{2}} = 4^{\frac{1}{3} \times \frac{1}{2}} = 4^{\frac{1}{6}}$.

Choice 4: $\sqrt{4}$

• This expression is equivalent to: $4^{\frac{1}{2}}$.

Now, let's compare the powers of 4:

• Choice 1: $4^{\frac{1}{12}}$
• Choice 2: $4^{\frac{1}{6}}$
• Choice 3: $4^{\frac{1}{6}}$
• Choice 4: $4^{\frac{1}{2}}$ - The largest exponent

The largest value among the given choices occurs when the exponent applied to the base 4 is maximized. Thus, the largest value is $\sqrt{4}$.