The descending order of `root(3)(4),sqrt2,root(6)(3)` and `root(4)(5)` is

Options

A. `root(3)(4)gtroot(4)(5)gtsqrt2gtroot(6)(3)`
B. `root(4)(5)gtroot(3)(4)gtroot(6)(3)gtsqrt2`
C. `sqrt2,root(6)(3)gtgtroot(3)(4)gtroot(4)(5)`
D. `root(6)(3)gtroot(4)5)gtroot(3)(4)gtsqrt2`

Answer

To find the descending order of the numbers \( \sqrt[3]{4}, \sqrt{2}, \sqrt[6]{3}, \text{ and } \sqrt[4]{5} \), we will convert each of these roots to a common exponent so we can compare them easily.

### Step-by-Step Solution:

1. **Convert each root to exponent form:**
- \( \sqrt[3]{4} = 4^{1/3} \)
- \( \sqrt{2} = 2^{1/2} \)
- \( \sqrt[6]{3} = 3^{1/6} \)
- \( \sqrt[4]{5} = 5^{1/4} \)

2. **Find the least common multiple (LCM) of the denominators:**
- The denominators are 3, 2, 6, and 4.
- The LCM of these numbers is 12.

3. **Rewrite each expression with a denominator of 12:**
- For \( \sqrt[3]{4} = 4^{1/3} \):
\[ 4^{1/3} = 4^{4/12} = 4^4 = 256^{1/12} \]
- For \( \sqrt{2} = 2^{1/2} \):
\[ 2^{1/2} = 2^{6/12} = 64^{1/12} \]
- For \( \sqrt[6]{3} = 3^{1/6} \):
\[ 3^{1/6} = 3^{2/12} = 9^{1/12} \]
- For \( \sqrt[4]{5} = 5^{1/4} \):
\[ 5^{1/4} = 5^{3/12} = 125^{1/12} \]

4. **Now we have the following expressions:**
- \( \sqrt[3]{4} = 256^{1/12} \)
- \( \sqrt{2} = 64^{1/12} \)
- \( \sqrt[6]{3} = 9^{1/12} \)
- \( \sqrt[4]{5} = 125^{1/12} \)

5. **Compare the bases:**
- The bases are 256, 125, 64, and 9.
- In descending order, these values are:
- 256 (from \( \sqrt[3]{4} \))
- 125 (from \( \sqrt[4]{5} \))
- 64 (from \( \sqrt{2} \))
- 9 (from \( \sqrt[6]{3} \))

6. **Write the final descending order:**
- Therefore, the descending order is:
\[ \sqrt[3]{4}, \sqrt[4]{5}, \sqrt{2}, \sqrt[6]{3} \]

### Final Answer:
The descending order is \( \sqrt[3]{4}, \sqrt[4]{5}, \sqrt{2}, \sqrt[6]{3} \).