The greatest one of `sqrt(2)`, `root3(3)`, `root6(6)`, `root5(5)` is

Options

A. `sqrt(2)`
B. `root3(3)`
C. `root6(6)`
D. `root5(5)`

Answer

To find the greatest among the numbers \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt[6]{6} \), and \( \sqrt[5]{5} \), we can express each of these roots in terms of powers with a common denominator. Here’s how we can do it step by step:

### Step-by-Step Solution:

1. **Convert each root to exponential form:**
- \( \sqrt{2} = 2^{1/2} \)
- \( \sqrt[3]{3} = 3^{1/3} \)
- \( \sqrt[6]{6} = 6^{1/6} \)
- \( \sqrt[5]{5} = 5^{1/5} \)

2. **Identify the denominators of the exponents:**
- The denominators are \( 2, 3, 6, \) and \( 5 \).

3. **Find the Least Common Multiple (LCM) of the denominators:**
- The LCM of \( 2, 3, 6, \) and \( 5 \) is \( 30 \).

4. **Rewrite each expression with the common denominator:**
- For \( \sqrt{2} \):
\[ 2^{1/2} = 2^{15/30} \]
- For \( \sqrt[3]{3} \):
\[ 3^{1/3} = 3^{10/30} \]
- For \( \sqrt[6]{6} \):
\[ 6^{1/6} = 6^{5/30} \]
- For \( \sqrt[5]{5} \):
\[ 5^{1/5} = 5^{6/30} \]

5. **Now we can compare the values:**
- \( 2^{15/30} \)
- \( 3^{10/30} \)
- \( 6^{5/30} \)
- \( 5^{6/30} \)

6. **Calculate the approximate values:**
- \( 2^{15/30} = \sqrt{2^{15}} \approx 5.656854 \)
- \( 3^{10/30} = \sqrt[3]{3^{10}} \approx 3.73719 \)
- \( 6^{5/30} = \sqrt[6]{6^{5}} \approx 2.689 \)
- \( 5^{6/30} = \sqrt[5]{5^{6}} \approx 3.659 \)

7. **Determine the greatest value:**
- Comparing the calculated values, we find:
- \( 5.656854 \) (from \( \sqrt{2} \)) is the largest.

### Conclusion:
The greatest among \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt[6]{6} \), and \( \sqrt[5]{5} \) is \( \sqrt{2} \).