The greatest among
`sqrt(7) - sqrt(5) , sqrt(5) - sqrt(3) , sqrt(9) - sqrt(7) , sqrt(11) - sqrt(9)` is

Options

A. `sqrt(7) - sqrt(5)`
B. `sqrt(5) - sqrt(3)`
C. `(sqrt(9)- sqrt(7)`
D. `sqrt(11) - sqrt(9)`

Answer

To find the greatest among the following expressions:

1. \( \sqrt{7} - \sqrt{5} \)
2. \( \sqrt{5} - \sqrt{3} \)
3. \( \sqrt{9} - \sqrt{7} \)
4. \( \sqrt{11} - \sqrt{9} \)

we will rationalize each expression and compare their values.

### Step 1: Rationalize \( \sqrt{7} - \sqrt{5} \)

To rationalize \( \sqrt{7} - \sqrt{5} \), we multiply and divide by \( \sqrt{7} + \sqrt{5} \):

\[ \sqrt{7} - \sqrt{5} = \frac{(\sqrt{7} - \sqrt{5})(\sqrt{7} + \sqrt{5})}{\sqrt{7} + \sqrt{5}} = \frac{7 - 5}{\sqrt{7} + \sqrt{5}} = \frac{2}{\sqrt{7} + \sqrt{5}} \]

### Step 2: Rationalize \( \sqrt{5} - \sqrt{3} \)

Now, we rationalize \( \sqrt{5} - \sqrt{3} \):

\[ \sqrt{5} - \sqrt{3} = \frac{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}{\sqrt{5} + \sqrt{3}} = \frac{5 - 3}{\sqrt{5} + \sqrt{3}} = \frac{2}{\sqrt{5} + \sqrt{3}} \]

### Step 3: Rationalize \( \sqrt{9} - \sqrt{7} \)

Next, we rationalize \( \sqrt{9} - \sqrt{7} \):

\[ \sqrt{9} - \sqrt{7} = \frac{(\sqrt{9} - \sqrt{7})(\sqrt{9} + \sqrt{7})}{\sqrt{9} + \sqrt{7}} = \frac{9 - 7}{\sqrt{9} + \sqrt{7}} = \frac{2}{\sqrt{9} + \sqrt{7}} \]

### Step 4: Rationalize \( \sqrt{11} - \sqrt{9} \)

Finally, we rationalize \( \sqrt{11} - \sqrt{9} \):

\[ \sqrt{11} - \sqrt{9} = \frac{(\sqrt{11} - \sqrt{9})(\sqrt{11} + \sqrt{9})}{\sqrt{11} + \sqrt{9}} = \frac{11 - 9}{\sqrt{11} + \sqrt{9}} = \frac{2}{\sqrt{11} + \sqrt{9}} \]

### Step 5: Compare the values

Now we have:

1. \( \sqrt{7} - \sqrt{5} = \frac{2}{\sqrt{7} + \sqrt{5}} \)
2. \( \sqrt{5} - \sqrt{3} = \frac{2}{\sqrt{5} + \sqrt{3}} \)
3. \( \sqrt{9} - \sqrt{7} = \frac{2}{\sqrt{9} + \sqrt{7}} \)
4. \( \sqrt{11} - \sqrt{9} = \frac{2}{\sqrt{11} + \sqrt{9}} \)

Since all numerators are equal (2), the greatest value will depend on the smallest denominator.

### Step 6: Evaluate the denominators

- \( \sqrt{7} + \sqrt{5} \)
- \( \sqrt{5} + \sqrt{3} \)
- \( \sqrt{9} + \sqrt{7} \)
- \( \sqrt{11} + \sqrt{9} \)

Calculating approximate values:

1. \( \sqrt{7} \approx 2.645 \) and \( \sqrt{5} \approx 2.236 \) gives \( \sqrt{7} + \sqrt{5} \approx 4.881 \)
2. \( \sqrt{5} \approx 2.236 \) and \( \sqrt{3} \approx 1.732 \) gives \( \sqrt{5} + \sqrt{3} \approx 3.968 \)
3. \( \sqrt{9} = 3 \) and \( \sqrt{7} \approx 2.645 \) gives \( \sqrt{9} + \sqrt{7} \approx 5.645 \)
4. \( \sqrt{11} \approx 3.317 \) and \( \sqrt{9} = 3 \) gives \( \sqrt{11} + \sqrt{9} \approx 6.317 \)

### Conclusion

The smallest denominator is \( \sqrt{5} + \sqrt{3} \), which corresponds to \( \sqrt{5} - \sqrt{3} \).

Thus, the greatest among the given expressions is:

\[ \boxed{\sqrt{5} - \sqrt{3}} \]