Arrange the following in the descreasing order ?
`(sqrt(23)-sqrt(21))`, `(sqrt(19)-sqrt(17))`, `(sqrt(21)-sqrt(19))`

Options

A. `(sqrt(23)-sqrt(21)) gt (sqrt(21)-sqrt(19)) gt (sqrt(19)-sqrt(17))`
B. `(sqrt(23)-sqrt(21)) gt (sqrt(19)-sqrt(17)) gt (sqrt(21)-sqrt(19))`
C. `(sqrt(19)-sqrt(17)) gt (sqrt(21)-sqrt(19)) gt (sqrt(23)-sqrt(21))`
D. `(sqrt(21)-sqrt(19)) gt (sqrt(23)-sqrt(21)) gt (sqrt(19)-sqrt(17))`

Answer

To arrange the terms \((\sqrt{23} - \sqrt{21})\), \((\sqrt{19} - \sqrt{17})\), and \((\sqrt{21} - \sqrt{19})\) in decreasing order, we can follow these steps:

### Step 1: Square each term
We will square each term to compare their values more easily.

1. **For \((\sqrt{23} - \sqrt{21})\)**:
\[ (\sqrt{23} - \sqrt{21})^2 = 23 + 21 - 2\sqrt{23 \cdot 21} = 44 - 2\sqrt{483} \]

2. **For \((\sqrt{19} - \sqrt{17})\)**:
\[ (\sqrt{19} - \sqrt{17})^2 = 19 + 17 - 2\sqrt{19 \cdot 17} = 36 - 2\sqrt{323} \]

3. **For \((\sqrt{21} - \sqrt{19})\)**:
\[ (\sqrt{21} - \sqrt{19})^2 = 21 + 19 - 2\sqrt{21 \cdot 19} = 40 - 2\sqrt{399} \]

### Step 2: Compare the squared values
Now we will compare the squared values of the three terms:

- **Term 1**: \(44 - 2\sqrt{483}\)
- **Term 2**: \(36 - 2\sqrt{323}\)
- **Term 3**: \(40 - 2\sqrt{399}\)

### Step 3: Estimate the square roots
To compare these values, we can estimate the square roots:

1. **Estimate \(\sqrt{483}\)**:
\(\sqrt{483} \approx 22\) (since \(22^2 = 484\))

2. **Estimate \(\sqrt{323}\)**:
\(\sqrt{323} \approx 18\) (since \(18^2 = 324\))

3. **Estimate \(\sqrt{399}\)**:
\(\sqrt{399} \approx 20\) (since \(20^2 = 400\))

### Step 4: Substitute estimates back into the expressions
Now substitute these estimates back into the expressions:

1. **For Term 1**:
\[ 44 - 2 \times 22 = 44 - 44 = 0 \]

2. **For Term 2**:
\[ 36 - 2 \times 18 = 36 - 36 = 0 \]

3. **For Term 3**:
\[ 40 - 2 \times 20 = 40 - 40 = 0 \]

### Step 5: Re-evaluate using a more precise approach
Since all terms seem to yield similar results, we can use a more precise approach to compare them directly without squaring.

### Step 6: Direct Comparison
1. **Compare \((\sqrt{23} - \sqrt{21})\) and \((\sqrt{21} - \sqrt{19})\)**:
\[ \sqrt{23} - \sqrt{21} > \sqrt{21} - \sqrt{19} \]
This indicates that \((\sqrt{23} - \sqrt{21})\) is greater.

2. **Compare \((\sqrt{21} - \sqrt{19})\) and \((\sqrt{19} - \sqrt{17})\)**:
\[ \sqrt{21} - \sqrt{19} > \sqrt{19} - \sqrt{17} \]
This indicates that \((\sqrt{21} - \sqrt{19})\) is greater.

### Final Arrangement
Thus, the final arrangement in decreasing order is:
\[ (\sqrt{23} - \sqrt{21}) > (\sqrt{21} - \sqrt{19}) > (\sqrt{19} - \sqrt{17}) \]