Arranging in Descending Order:

To arrange the given numbers in descending order, we need to compare their values and place them from the highest to the lowest value. Let's compare the given numbers step by step.

Step 1: Compare 6√7 and 12√3:
To compare these two numbers, we can calculate their decimal approximations.

Calculating the decimal approximation of 6√7:
6√7 ≈ 6 × 2.65 ≈ 15.9

Calculating the decimal approximation of 12√3:
12√3 ≈ 12 × 1.73 ≈ 20.76

Since 20.76 is greater than 15.9, we can conclude that 12√3 is greater than 6√7.

Step 2: Compare 4√5 and 12√3:
To compare these two numbers, we can calculate their decimal approximations.

Calculating the decimal approximation of 4√5:
4√5 ≈ 4 × 2.24 ≈ 8.96

Calculating the decimal approximation of 12√3:
12√3 ≈ 12 × 1.73 ≈ 20.76

Since 20.76 is greater than 8.96, we can conclude that 12√3 is greater than 4√5.

Step 3: Compare 3√2 and 4√5:
To compare these two numbers, we can calculate their decimal approximations.

Calculating the decimal approximation of 3√2:
3√2 ≈ 3 × 1.41 ≈ 4.23

Calculating the decimal approximation of 4√5:
4√5 ≈ 4 × 2.24 ≈ 8.96

Since 8.96 is greater than 4.23, we can conclude that 4√5 is greater than 3√2.

Step 4: Compare 3√2 and 6√7:
To compare these two numbers, we can calculate their decimal approximations.

Calculating the decimal approximation of 3√2:
3√2 ≈ 3 × 1.41 ≈ 4.23

Calculating the decimal approximation of 6√7:
6√7 ≈ 6 × 2.65 ≈ 15.9

Since 15.9 is greater than 4.23, we can conclude that 6√7 is greater than 3√2.

Final Order:

Based on the comparisons made above, we can conclude that the numbers arranged in descending order are:

12√3 > 6√7 > 4√5 > 3√2

Therefore, the descending order of the given numbers is 12√3, 6√7, 4√5, and 3√2.