Arrange the following in the decending order,
`5^(1//4),4^(1//3),6^(1//5)`

Options

A. `4^(1//3),5^(1//4),6^(1//5)`
B. `5^(1//4),4^(1//3), 6^(1//5)`
C. `6^(1//5),4^(1//3), 5^(1//4)`
D. `5^(1//4),4^(1//3), 6^(1//5)`

Answer

To arrange the numbers \(5^{1/4}\), \(4^{1/3}\), and \(6^{1/5}\) in descending order, we can follow these steps:

### Step 1: Convert the expressions to a common base
To compare these numbers effectively, we can take the logarithm of each expression. This will allow us to compare their values without calculating the actual roots.

### Step 2: Calculate the logarithms
1. For \(5^{1/4}\):
\[ \log(5^{1/4}) = \frac{1}{4} \log(5) \]

2. For \(4^{1/3}\):
\[ \log(4^{1/3}) = \frac{1}{3} \log(4) \]

3. For \(6^{1/5}\):
\[ \log(6^{1/5}) = \frac{1}{5} \log(6) \]

### Step 3: Use approximate values for logarithms
Using approximate values:
- \(\log(5) \approx 0.699\)
- \(\log(4) \approx 0.602\)
- \(\log(6) \approx 0.778\)

Now we can calculate:

1. \(5^{1/4}\):
\[ \frac{1}{4} \log(5) \approx \frac{1}{4} \times 0.699 \approx 0.17475 \]

2. \(4^{1/3}\):
\[ \frac{1}{3} \log(4) \approx \frac{1}{3} \times 0.602 \approx 0.20067 \]

3. \(6^{1/5}\):
\[ \frac{1}{5} \log(6) \approx \frac{1}{5} \times 0.778 \approx 0.1556 \]

### Step 4: Compare the calculated values
Now we compare the values:
- \(5^{1/4} \approx 0.17475\)
- \(4^{1/3} \approx 0.20067\)
- \(6^{1/5} \approx 0.1556\)

### Step 5: Arrange in descending order
From the comparisons:
- \(4^{1/3} > 5^{1/4} > 6^{1/5}\)

Thus, the descending order is:
\[ 4^{1/3}, 5^{1/4}, 6^{1/5} \]

### Final Answer
The final answer in descending order is:
\[ 4^{1/3}, 5^{1/4}, 6^{1/5} \]

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