Arrange the numbers in descending order: `3sqrt""5,2sqrt""8,4sqrt""6`

Options

A. `4sqrt""6lt3sqrt""5lt2sqrt""8`
B. `2sqrt""8gt4sqrt""6gt3sqrt""5`
C. `4sqrt""6gt3sqrt""5gt2sqrt""8`
D. `2sqrt""8gt3sqrt""5lt4sqrt""6`

Answer

To arrange the numbers \(3\sqrt{5}\), \(2\sqrt{8}\), and \(4\sqrt{6}\) in descending order, we will first convert each expression into a simpler form that makes it easier to compare their values.

### Step 1: Simplify each expression

1. **For \(3\sqrt{5}\)**:
\[ 3\sqrt{5} = \sqrt{(3^2) \cdot 5} = \sqrt{9 \cdot 5} = \sqrt{45} \]

2. **For \(2\sqrt{8}\)**:
\[ 2\sqrt{8} = 2\sqrt{(4 \cdot 2)} = 2 \cdot 2\sqrt{2} = 4\sqrt{2} \]
We can also express it as:
\[ 2\sqrt{8} = \sqrt{(2^2) \cdot 8} = \sqrt{4 \cdot 8} = \sqrt{32} \]

3. **For \(4\sqrt{6}\)**:
\[ 4\sqrt{6} = \sqrt{(4^2) \cdot 6} = \sqrt{16 \cdot 6} = \sqrt{96} \]

### Step 2: Compare the simplified values

Now we have:
- \(3\sqrt{5} = \sqrt{45}\)
- \(2\sqrt{8} = \sqrt{32}\)
- \(4\sqrt{6} = \sqrt{96}\)

### Step 3: Determine the order

To compare \(\sqrt{45}\), \(\sqrt{32}\), and \(\sqrt{96}\), we can compare the values under the square roots:

1. **Calculate approximate values**:
- \(\sqrt{45} \approx 6.71\)
- \(\sqrt{32} \approx 5.66\)
- \(\sqrt{96} \approx 9.80\)

From these calculations, we can see that:
\[ \sqrt{96} > \sqrt{45} > \sqrt{32} \]

### Step 4: Write the final order

Thus, in descending order, we have:
\[ 4\sqrt{6} > 3\sqrt{5} > 2\sqrt{8} \]

### Conclusion

The final arrangement in descending order is:
\[ 4\sqrt{6}, 3\sqrt{5}, 2\sqrt{8} \]