Number Comparison: Determining the Largest Value in a Set

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

• Step 1: Convert each expression into exponential form.
• Step 2: Compare the resulting expressions by examining the exponents.
• Step 3: Identify which expression corresponds to the largest value.

Let's work through the solution:

Step 1: Convert each root to exponential form:
- $\sqrt{2} = 2^{0.5}$
- $\sqrt[3]{2} = 2^{1/3} \approx 2^{0.333}$
- $\sqrt[4]{2} = 2^{1/4} \approx 2^{0.25}$
- $\sqrt[5]{2} = 2^{1/5} \approx 2^{0.2}$

Step 2: Compare the exponents $0.5$, $0.333$, $0.25$, and $0.2$. Clearly, $0.5$ is the largest among these values.

Step 3: The expression with the largest exponent is $\sqrt{2} = 2^{0.5}$, so $\sqrt{2}$ is the largest value.

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