Which of the following is the largest number ?
`sqrt(2),root3(3), root4(4), root6(6)`

Options

A. `sqrt(2)`
B. `root3(3)`
C. `root4(4)`
D. `root6(6)`

Answer

To determine which of the following numbers is the largest: \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt[4]{4} \), or \( \sqrt[6]{6} \), we can compare them by raising each number to a common power. The least common multiple (LCM) of the indices (2, 3, 4, and 6) will help us with this comparison.

### Step-by-Step Solution:

1. **Identify the numbers and their roots**:
- \( \sqrt{2} = 2^{1/2} \)
- \( \sqrt[3]{3} = 3^{1/3} \)
- \( \sqrt[4]{4} = 4^{1/4} = (2^2)^{1/4} = 2^{1/2} = \sqrt{2} \)
- \( \sqrt[6]{6} = 6^{1/6} \)

2. **Find the LCM of the indices**:
- The indices are 2, 3, 4, and 6. The LCM of these numbers is 12.

3. **Raise each number to the power of 12**:
- \( \sqrt{2}^{12} = (2^{1/2})^{12} = 2^{12/2} = 2^6 = 64 \)
- \( \sqrt[3]{3}^{12} = (3^{1/3})^{12} = 3^{12/3} = 3^4 = 81 \)
- \( \sqrt[4]{4}^{12} = (4^{1/4})^{12} = (2^2)^{12/4} = 2^6 = 64 \)
- \( \sqrt[6]{6}^{12} = (6^{1/6})^{12} = 6^{12/6} = 6^2 = 36 \)

4. **Compare the results**:
- \( \sqrt{2}^{12} = 64 \)
- \( \sqrt[3]{3}^{12} = 81 \)
- \( \sqrt[4]{4}^{12} = 64 \)
- \( \sqrt[6]{6}^{12} = 36 \)

5. **Determine the largest value**:
- Among the calculated values, \( 81 \) (from \( \sqrt[3]{3} \)) is the largest.

### Conclusion:
Thus, the largest number among \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt[4]{4} \), and \( \sqrt[6]{6} \) is \( \sqrt[3]{3} \).