Which of the following number is the greatest ?
2^(1/2), 3^(1/3), 4^(1/4), 6^(1/6

Options

A. `sqrt2`
B. `3^(1/3)
C. 4^(1/4)
D. 6^(1/6

Answer

To determine which of the following numbers is the greatest: \(2^{1/2}\), \(3^{1/3}\), \(4^{1/4}\), and \(6^{1/6}\), we can compare these numbers by rewriting them in a common form.

### Step-by-Step Solution:

1. **Rewrite the Numbers:**
- \(2^{1/2} = \sqrt{2}\)
- \(3^{1/3} = \text{Cube root of } 3\)
- \(4^{1/4} = \text{Fourth root of } 4\)
- \(6^{1/6} = \text{Sixth root of } 6\)

2. **Convert to Exponential Form:**
We can express each term with a common exponent by finding the least common multiple (LCM) of the denominators of the exponents (2, 3, 4, and 6). The LCM of these numbers is 12.

3. **Express Each Term with LCM as the Denominator:**
- For \(2^{1/2}\):
\[ 2^{1/2} = 2^{6/12} = (2^6)^{1/12} = 64^{1/12} \]
- For \(3^{1/3}\):
\[ 3^{1/3} = 3^{4/12} = (3^4)^{1/12} = 81^{1/12} \]
- For \(4^{1/4}\):
\[ 4^{1/4} = 4^{3/12} = (4^3)^{1/12} = 64^{1/12} \]
- For \(6^{1/6}\):
\[ 6^{1/6} = 6^{2/12} = (6^2)^{1/12} = 36^{1/12} \]

4. **Compare the Values:**
Now we have:
- \(2^{1/2} = 64^{1/12}\)
- \(3^{1/3} = 81^{1/12}\)
- \(4^{1/4} = 64^{1/12}\)
- \(6^{1/6} = 36^{1/12}\)

Since all terms are raised to the same power (1/12), we can compare the bases:
- \(64\)
- \(81\)
- \(64\)
- \(36\)

5. **Identify the Greatest Value:**
The greatest base among \(64\), \(81\), and \(36\) is \(81\). Therefore, the greatest number is:
\[ 3^{1/3} \]

### Conclusion:
The greatest number among \(2^{1/2}\), \(3^{1/3}\), \(4^{1/4}\), and \(6^{1/6}\) is \(3^{1/3}\).