`(1)/(sqrt(9)-sqrt(8))-(1)/(sqrt(8)-sqrt(7))+(1)/(sqrt(7)-sqrt(6))-(1)/(sqrt(6)-sqrt(5))+(1)/(sqrt(5)-sqrt(4))=?`

Options

A. 0
B. 1
C. `1/3`
D. 5

Answer

To solve the expression

\[ \frac{1}{\sqrt{9} - \sqrt{8}} - \frac{1}{\sqrt{8} - \sqrt{7}} + \frac{1}{\sqrt{7} - \sqrt{6}} - \frac{1}{\sqrt{6} - \sqrt{5}} + \frac{1}{\sqrt{5} - \sqrt{4}}, \]

we will rationalize each term.

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

To rationalize, multiply the numerator and denominator by \(\sqrt{9} + \sqrt{8}\):

\[ \frac{1}{\sqrt{9} - \sqrt{8}} \cdot \frac{\sqrt{9} + \sqrt{8}}{\sqrt{9} + \sqrt{8}} = \frac{\sqrt{9} + \sqrt{8}}{(\sqrt{9})^2 - (\sqrt{8})^2} = \frac{\sqrt{9} + \sqrt{8}}{9 - 8} = \sqrt{9} + \sqrt{8} = 3 + \sqrt{8}. \]

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

Multiply the numerator and denominator by \(\sqrt{8} + \sqrt{7}\):

\[ -\frac{1}{\sqrt{8} - \sqrt{7}} \cdot \frac{\sqrt{8} + \sqrt{7}}{\sqrt{8} + \sqrt{7}} = -\frac{\sqrt{8} + \sqrt{7}}{(\sqrt{8})^2 - (\sqrt{7})^2} = -\frac{\sqrt{8} + \sqrt{7}}{8 - 7} = -(\sqrt{8} + \sqrt{7}). \]

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

Multiply the numerator and denominator by \(\sqrt{7} + \sqrt{6}\):

\[ \frac{1}{\sqrt{7} - \sqrt{6}} \cdot \frac{\sqrt{7} + \sqrt{6}}{\sqrt{7} + \sqrt{6}} = \frac{\sqrt{7} + \sqrt{6}}{(\sqrt{7})^2 - (\sqrt{6})^2} = \frac{\sqrt{7} + \sqrt{6}}{7 - 6} = \sqrt{7} + \sqrt{6}. \]

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

Multiply the numerator and denominator by \(\sqrt{6} + \sqrt{5}\):

\[ -\frac{1}{\sqrt{6} - \sqrt{5}} \cdot \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} + \sqrt{5}} = -\frac{\sqrt{6} + \sqrt{5}}{(\sqrt{6})^2 - (\sqrt{5})^2} = -\frac{\sqrt{6} + \sqrt{5}}{6 - 5} = -(\sqrt{6} + \sqrt{5}). \]

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

Multiply the numerator and denominator by \(\sqrt{5} + \sqrt{4}\):

\[ \frac{1}{\sqrt{5} - \sqrt{4}} \cdot \frac{\sqrt{5} + \sqrt{4}}{\sqrt{5} + \sqrt{4}} = \frac{\sqrt{5} + \sqrt{4}}{(\sqrt{5})^2 - (\sqrt{4})^2} = \frac{\sqrt{5} + \sqrt{4}}{5 - 4} = \sqrt{5} + \sqrt{4}. \]

### Step 6: Combine all terms

Now we combine all the rationalized terms:

\[ (3 + \sqrt{8}) - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + 2). \]

Notice that \(\sqrt{8}\) cancels with \(-\sqrt{8}\), \(\sqrt{7}\) cancels with \(-\sqrt{7}\), and \(\sqrt{6}\) cancels with \(-\sqrt{6}\). Thus, we are left with:

\[ 3 + 2 = 5. \]

### Final Answer

The value of the expression is

\[ \boxed{5}. \]