Write in descending order :
`2root(4)6 ` and `3 root(4)2`

Answer

To solve the problem of writing the numbers \(2 \cdot \sqrt[4]{6}\) and \(3 \cdot \sqrt[4]{2}\) in descending order, we can follow these steps:

### Step 1: Rewrite the numbers in terms of the fourth root
We want to express both numbers in a way that allows us to compare them easily. We can rewrite \(2\) and \(3\) in terms of the fourth root.

- For \(2 \cdot \sqrt[4]{6}\):
\[ 2 = \sqrt[4]{2^4} \quad \text{(since } 2 = 2^1 \text{ and } 2^4 = 16 \text{)} \]
Therefore,
\[ 2 \cdot \sqrt[4]{6} = \sqrt[4]{2^4} \cdot \sqrt[4]{6} = \sqrt[4]{2^4 \cdot 6} = \sqrt[4]{96} \]

- For \(3 \cdot \sqrt[4]{2}\):
\[ 3 = \sqrt[4]{3^4} \quad \text{(since } 3 = 3^1 \text{ and } 3^4 = 81 \text{)} \]
Therefore,
\[ 3 \cdot \sqrt[4]{2} = \sqrt[4]{3^4} \cdot \sqrt[4]{2} = \sqrt[4]{3^4 \cdot 2} = \sqrt[4]{162} \]

### Step 2: Compare the two fourth roots
Now we have:
- \(2 \cdot \sqrt[4]{6} = \sqrt[4]{96}\)
- \(3 \cdot \sqrt[4]{2} = \sqrt[4]{162}\)

To compare \(\sqrt[4]{96}\) and \(\sqrt[4]{162}\), we can compare the numbers inside the fourth roots:
- \(96\) and \(162\)

Since \(162 > 96\), we conclude that:
\[ \sqrt[4]{162} > \sqrt[4]{96} \]

### Step 3: Write in descending order
Now that we have established that \(3 \cdot \sqrt[4]{2}\) is greater than \(2 \cdot \sqrt[4]{6}\), we can write the numbers in descending order:
\[ 3 \cdot \sqrt[4]{2} > 2 \cdot \sqrt[4]{6} \]

Thus, the final answer in descending order is:
\[ 3 \cdot \sqrt[4]{2}, \quad 2 \cdot \sqrt[4]{6} \]