Write `4sqrt(6), sqrt(2), 3sqrt(4)` in ascending order.

Answer

To arrange the numbers \(4\sqrt{6}\), \(\sqrt{2}\), and \(3\sqrt{4}\) in ascending order, we will first simplify each expression and then compare their values.

### Step 1: Simplify each expression

1. **Simplifying \(4\sqrt{6}\)**:
- This expression is already in its simplest form. We will keep it as \(4\sqrt{6}\).

2. **Simplifying \(\sqrt{2}\)**:
- This expression is also in its simplest form. We will keep it as \(\sqrt{2}\).

3. **Simplifying \(3\sqrt{4}\)**:
- We know that \(\sqrt{4} = 2\).
- Therefore, \(3\sqrt{4} = 3 \times 2 = 6\).

### Step 2: Compare the values

Now we have:
- \(4\sqrt{6}\)
- \(\sqrt{2}\)
- \(6\)

Next, we will calculate the approximate decimal values to compare them easily:
- \(4\sqrt{6} \approx 4 \times 2.45 \approx 9.8\)
- \(\sqrt{2} \approx 1.41\)
- \(6\) is already a whole number.

### Step 3: Arrange in ascending order

Now we can compare:
- \(\sqrt{2} \approx 1.41\) (smallest)
- \(6\) (next)
- \(4\sqrt{6} \approx 9.8\) (largest)

Thus, the ascending order is:
\[ \sqrt{2}, \quad 3\sqrt{4}, \quad 4\sqrt{6} \]

### Final Answer:
The numbers in ascending order are:
\[ \sqrt{2}, \quad 3\sqrt{4}, \quad 4\sqrt{6} \]