The value of `[1/(sqrt9-sqrt8)]-[1/(sqrt8-sqrt7)]+[1/(sqrt7-sqrt6)]-[1/(sqrt6-sqrt5)]+[1/(sqrt5-sqrt4)]` is
A)6
B)5
C)-7
D)-6

Options

A. 6
B. 5
C. `-7`
D. `-6`

Answer

To solve the expression

\[ E = \left[\frac{1}{\sqrt{9} - \sqrt{8}}\right] - \left[\frac{1}{\sqrt{8} - \sqrt{7}}\right] + \left[\frac{1}{\sqrt{7} - \sqrt{6}}\right] - \left[\frac{1}{\sqrt{6} - \sqrt{5}}\right] + \left[\frac{1}{\sqrt{5} - \sqrt{4}}\right] \]

we will rationalize each term in the expression.

### Step 1: Rationalize the first term \(\frac{1}{\sqrt{9} - \sqrt{8}}\)

To rationalize, multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{1}{\sqrt{9} - \sqrt{8}} \cdot \frac{\sqrt{9} + \sqrt{8}}{\sqrt{9} + \sqrt{8}} = \frac{\sqrt{9} + \sqrt{8}}{(\sqrt{9})^2 - (\sqrt{8})^2} = \frac{\sqrt{9} + \sqrt{8}}{9 - 8} = \sqrt{9} + \sqrt{8} \]

### Step 2: Rationalize the second term \(-\frac{1}{\sqrt{8} - \sqrt{7}}\)

Similarly, we rationalize:

\[ -\frac{1}{\sqrt{8} - \sqrt{7}} \cdot \frac{\sqrt{8} + \sqrt{7}}{\sqrt{8} + \sqrt{7}} = -\frac{\sqrt{8} + \sqrt{7}}{8 - 7} = -(\sqrt{8} + \sqrt{7}) \]

### Step 3: Rationalize the third term \(\frac{1}{\sqrt{7} - \sqrt{6}}\)

Rationalizing gives:

\[ \frac{1}{\sqrt{7} - \sqrt{6}} \cdot \frac{\sqrt{7} + \sqrt{6}}{\sqrt{7} + \sqrt{6}} = \frac{\sqrt{7} + \sqrt{6}}{7 - 6} = \sqrt{7} + \sqrt{6} \]

### Step 4: Rationalize the fourth term \(-\frac{1}{\sqrt{6} - \sqrt{5}}\)

Rationalizing gives:

\[ -\frac{1}{\sqrt{6} - \sqrt{5}} \cdot \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} + \sqrt{5}} = -\frac{\sqrt{6} + \sqrt{5}}{6 - 5} = -(\sqrt{6} + \sqrt{5}) \]

### Step 5: Rationalize the fifth term \(\frac{1}{\sqrt{5} - \sqrt{4}}\)

Rationalizing gives:

\[ \frac{1}{\sqrt{5} - \sqrt{4}} \cdot \frac{\sqrt{5} + \sqrt{4}}{\sqrt{5} + \sqrt{4}} = \frac{\sqrt{5} + \sqrt{4}}{5 - 4} = \sqrt{5} + \sqrt{4} \]

### Step 6: Combine all the terms

Now we combine all the rationalized terms:

\[ E = (\sqrt{9} + \sqrt{8}) - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + \sqrt{4}) \]

### Step 7: Simplify the expression

Notice that terms will cancel out:

- \(\sqrt{8}\) from the first and second terms cancels.
- \(\sqrt{7}\) from the second and third terms cancels.
- \(\sqrt{6}\) from the third and fourth terms cancels.
- \(\sqrt{5}\) from the fourth and fifth terms cancels.

This leaves us with:

\[ E = \sqrt{9} + \sqrt{4} = 3 + 2 = 5 \]

### Final Answer

Thus, the value of the expression is

\[ \boxed{5} \]