The greatest among the following numbers `(3)^(1/3),(2)^(1/2),1,(6)^(1/6)`is :

Options

A. `(2)^((1)/(2))`
B. 1
C. `(6)^((1)/(6))`
D. `(3)^((1)/(3))`

Answer

To find the greatest among the numbers \( (3)^{1/3}, (2)^{1/2}, 1, (6)^{1/6} \), we can compare them by expressing them with a common exponent. In this case, we will use \( \frac{1}{6} \) as the common exponent.

### Step-by-step Solution:

1. **Convert \( (3)^{1/3} \) to a power of \( \frac{1}{6} \)**:
\[ (3)^{1/3} = (3^{2})^{1/6} = 9^{1/6} \]
Here, we multiply the exponent \( \frac{1}{3} \) by \( \frac{2}{2} \) to convert it to \( \frac{1}{6} \).

**Hint**: To convert the exponent, multiply both the numerator and denominator by the same number.

2. **Convert \( (2)^{1/2} \) to a power of \( \frac{1}{6} \)**:
\[ (2)^{1/2} = (2^{3})^{1/6} = 8^{1/6} \]
Similarly, we multiply the exponent \( \frac{1}{2} \) by \( \frac{3}{3} \).

**Hint**: Use the same method of adjusting the exponent to match \( \frac{1}{6} \).

3. **Convert \( 1 \) to a power of \( \frac{1}{6} \)**:
\[ 1 = (1)^{1} = (1^{6})^{1/6} = 1^{1/6} \]
Here, we express \( 1 \) as \( 1^{6} \) to match the exponent.

**Hint**: Any number raised to any power is still that number if it's 1.

4. **Convert \( (6)^{1/6} \)**:
\[ (6)^{1/6} = (6)^{1/6} \]
This number is already in the desired form.

**Hint**: No conversion needed if the exponent is already \( \frac{1}{6} \).

5. **Compare the bases**:
Now we compare the bases:
- \( 9^{1/6} \)
- \( 8^{1/6} \)
- \( 1^{1/6} \)
- \( 6^{1/6} \)

The bases are \( 9, 8, 1, \) and \( 6 \). The greatest base is \( 9 \).

6. **Conclusion**:
Therefore, the greatest among the given numbers is:
\[ (3)^{1/3} \]

### Final Answer:
The greatest among the numbers \( (3)^{1/3}, (2)^{1/2}, 1, (6)^{1/6} \) is \( (3)^{1/3} \).