Which no. is the greatest number from the following:
`root(3)(4),root(6)(15),root(4)(6)` and `root(12)(245)`

Options

A. `root(3)(4)`
B. `root(4)(6)`
C. `root(6)(15)`
D. `root(12)(245)`

Answer

To find the greatest number among the given roots, we need to evaluate each expression: \( \sqrt[3]{4} \), \( \sqrt[6]{15} \), \( \sqrt[4]{6} \), and \( \sqrt[12]{245} \).

### Step-by-Step Solution:

1. **Rewrite the Roots in Exponential Form**:
- \( \sqrt[3]{4} = 4^{1/3} \)
- \( \sqrt[6]{15} = 15^{1/6} \)
- \( \sqrt[4]{6} = 6^{1/4} \)
- \( \sqrt[12]{245} = 245^{1/12} \)

2. **Find the Least Common Multiple (LCM) of the Denominators**:
- The denominators are 3, 6, 4, and 12.
- The LCM of these numbers is 12.

3. **Express Each Root with a Common Denominator**:
- Rewrite each expression using the LCM:
- \( 4^{1/3} = 4^{4/12} \)
- \( 15^{1/6} = 15^{2/12} \)
- \( 6^{1/4} = 6^{3/12} \)
- \( 245^{1/12} = 245^{1/12} \)

4. **Combine the Expressions Under the Same Root**:
- Now we can express them all as 12th roots:
- \( \sqrt[12]{4^4} = \sqrt[12]{256} \)
- \( \sqrt[12]{15^2} = \sqrt[12]{225} \)
- \( \sqrt[12]{6^3} = \sqrt[12]{216} \)
- \( \sqrt[12]{245} = \sqrt[12]{245} \)

5. **Calculate the Values**:
- \( 256 \) (from \( 4^4 \))
- \( 225 \) (from \( 15^2 \))
- \( 216 \) (from \( 6^3 \))
- \( 245 \) (from \( 245 \))

6. **Compare the Values**:
- Now we compare the values:
- \( 256 > 225 > 216 > 245 \)

7. **Identify the Greatest Value**:
- The greatest value is \( 256 \), which corresponds to \( \sqrt[3]{4} \).

### Conclusion:
The greatest number among the given options is \( \sqrt[3]{4} \).