Arrange the following in descending order : `root4(5)`, `root3(4)` and `root4(6)`

Options

A. `root4(6) gt root3(4) gt root4(5)`
B. `root4(6) gt root4(5) gt root3(4)`
C. `root3(4) gt root4(5) gt root4(6)`
D. `root3(4) gt root4(6) gt root4(5)`

Answer

To arrange the numbers \( \sqrt[4]{5} \), \( \sqrt[3]{4} \), and \( \sqrt[4]{6} \) in descending order, we can convert each root into an exponent form and then compare their values.

### Step-by-Step Solution:

1. **Convert each root to exponent form**:
- \( \sqrt[4]{5} = 5^{1/4} \)
- \( \sqrt[3]{4} = 4^{1/3} \)
- \( \sqrt[4]{6} = 6^{1/4} \)

2. **Find a common exponent to compare the values**:
- The least common multiple (LCM) of the denominators (4 and 3) is 12.
- Therefore, we can express each term with a power of 12:
- \( 5^{1/4} = 5^{3/12} \)
- \( 4^{1/3} = 4^{4/12} \)
- \( 6^{1/4} = 6^{3/12} \)

3. **Rewrite the expressions**:
- Now we can rewrite the expressions as:
- \( 5^{3/12} \)
- \( 4^{4/12} \)
- \( 6^{3/12} \)

4. **Calculate the values of each expression**:
- Calculate \( 5^{3} \), \( 4^{4} \), and \( 6^{3} \):
- \( 5^{3} = 125 \)
- \( 4^{4} = 256 \)
- \( 6^{3} = 216 \)

5. **Compare the calculated values**:
- Now we compare:
- \( 5^{3/12} \) corresponds to \( 125 \)
- \( 4^{4/12} \) corresponds to \( 256 \)
- \( 6^{3/12} \) corresponds to \( 216 \)

6. **Arrange in descending order**:
- From the calculated values, we can see:
- \( 4^{4/12} (256) > 6^{3/12} (216) > 5^{3/12} (125) \)
- Therefore, the descending order is:
- \( \sqrt[3]{4} > \sqrt[4]{6} > \sqrt[4]{5} \)

### Final Answer:
The arrangement in descending order is:
\[ \sqrt[3]{4}, \sqrt[4]{6}, \sqrt[4]{5} \]