If the numbers `root3(9)`, `root4(20)`, `root6(25)` are arranged in ascending order then the right arrangement is

Options

A. `root6(25) lt root4(20) lt root3(9)`
B. `root3(9) lt root4(20) lt root6(25)`
C. `root4(20) lt root6(25) lt root3(9)`
D. `root6(25) lt root3(9) lt root4(20)`

Answer

To solve the problem of arranging the numbers \( \sqrt[3]{9} \), \( \sqrt[4]{20} \), and \( \sqrt[6]{25} \) in ascending order, we will follow these steps:

### Step 1: Rewrite the roots in exponent form
We can express the roots in terms of exponents:
- \( \sqrt[3]{9} = 9^{1/3} \)
- \( \sqrt[4]{20} = 20^{1/4} \)
- \( \sqrt[6]{25} = 25^{1/6} \)

### Step 2: Find a common denominator for the exponents
To compare these numbers, we need to express them with a common exponent. The denominators of the exponents are 3, 4, and 6. The least common multiple (LCM) of these numbers is 12.

### Step 3: Rewrite each number with the common exponent
Now we will rewrite each number using the common denominator of 12:
- For \( \sqrt[3]{9} \):
\[ 9^{1/3} = 9^{4/12} = (9^4)^{1/12} \]
- For \( \sqrt[4]{20} \):
\[ 20^{1/4} = 20^{3/12} = (20^3)^{1/12} \]
- For \( \sqrt[6]{25} \):
\[ 25^{1/6} = 25^{2/12} = (25^2)^{1/12} \]

### Step 4: Calculate the values of the bases
Next, we need to calculate the values of \( 9^4 \), \( 20^3 \), and \( 25^2 \):
- \( 9^4 = 6561 \)
- \( 20^3 = 8000 \)
- \( 25^2 = 625 \)

### Step 5: Compare the values
Now we compare the values:
- \( 625^{1/12} \) (from \( \sqrt[6]{25} \))
- \( 6561^{1/12} \) (from \( \sqrt[3]{9} \))
- \( 8000^{1/12} \) (from \( \sqrt[4]{20} \))

Since \( 625 < 6561 < 8000 \), we can conclude that:
- \( \sqrt[6]{25} < \sqrt[3]{9} < \sqrt[4]{20} \)

### Step 6: Write the final arrangement in ascending order
Thus, the numbers in ascending order are:
\[ \sqrt[6]{25}, \sqrt[3]{9}, \sqrt[4]{20} \]

### Final Answer
The correct arrangement in ascending order is:
\[ \sqrt[6]{25}, \sqrt[3]{9}, \sqrt[4]{20} \]