Arrange `root(4)(3),root(6)(10),root(12)(25)` in descending order.

Options

A. `root(6)(10)gtroot(4)(3)gtroot(12)(25)`
B. `root(12)(25)gtroot(4)(3)gtroot(6)(10)`
C. `root(6)(10)gtroot(12)(25)gtroot(4)(3)`
D. `root(4)(3)gtroot(12)(25)gtroot(6)(10)`

Answer

To arrange the numbers \( \sqrt[4]{3}, \sqrt[6]{10}, \sqrt[12]{25} \) in descending order, we can follow these steps:

### Step 1: Convert to a Common Root
To compare these roots easily, we need to express all of them with a common root. The least common multiple (LCM) of the indices (4, 6, and 12) is 12. Therefore, we will convert each root to have a denominator of 12.

### Step 2: Convert \( \sqrt[4]{3} \)
To convert \( \sqrt[4]{3} \) to a 12th root:
\[ \sqrt[4]{3} = \sqrt[12]{3^{3}} = \sqrt[12]{27} \]

### Step 3: Convert \( \sqrt[6]{10} \)
To convert \( \sqrt[6]{10} \) to a 12th root:
\[ \sqrt[6]{10} = \sqrt[12]{10^{2}} = \sqrt[12]{100} \]

### Step 4: Convert \( \sqrt[12]{25} \)
This is already in the desired form:
\[ \sqrt[12]{25} = \sqrt[12]{25} \]

### Step 5: Compare the Values
Now we have:
- \( \sqrt[12]{27} \)
- \( \sqrt[12]{100} \)
- \( \sqrt[12]{25} \)

Next, we compare the values inside the roots:
- \( 27 \) (from \( \sqrt[12]{27} \))
- \( 100 \) (from \( \sqrt[12]{100} \))
- \( 25 \) (from \( \sqrt[12]{25} \))

### Step 6: Arrange in Descending Order
Now we can arrange these values in descending order based on their radicands:
- \( 100 \) (largest)
- \( 27 \) (middle)
- \( 25 \) (smallest)

Thus, the order of the original roots in descending order is:
\[ \sqrt[6]{10} > \sqrt[4]{3} > \sqrt[12]{25} \]

### Final Answer
The final arrangement in descending order is:
\[ \sqrt[6]{10} > \sqrt[4]{3} > \sqrt[12]{25} \]

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