The value of `(1)/(sqrt(4)+sqrt(5))+(1)/(sqrt(5)+sqrt(6))+(1)/(sqrt(6)+sqrt(7))+(1)/(sqrt(7)+sqrt(8))+(1)/(sqrt(8)+sqrt(9))` is

Answer

To solve the expression

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

we will simplify each term by rationalizing the denominators.

### Step 1: Rationalize the first term
We start with the first term:

\[ \frac{1}{\sqrt{4}+\sqrt{5}}. \]

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

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

### Step 2: Rationalize the second term
Now for the second term:

\[ \frac{1}{\sqrt{5}+\sqrt{6}}. \]

Using the same method:

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

### Step 3: Rationalize the third term
For the third term:

\[ \frac{1}{\sqrt{6}+\sqrt{7}}. \]

Rationalizing gives:

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

### Step 4: Rationalize the fourth term
For the fourth term:

\[ \frac{1}{\sqrt{7}+\sqrt{8}}. \]

Rationalizing gives:

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

### Step 5: Rationalize the fifth term
For the fifth term:

\[ \frac{1}{\sqrt{8}+\sqrt{9}}. \]

Rationalizing gives:

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

### Step 6: Combine all terms
Now we can combine all the rationalized terms:

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

Notice that most terms cancel out:

- \(\sqrt{5}\) cancels with \(-\sqrt{5}\),
- \(\sqrt{6}\) cancels with \(-\sqrt{6}\),
- \(\sqrt{7}\) cancels with \(-\sqrt{7}\),
- \(\sqrt{8}\) cancels with \(-\sqrt{8}\).

### Final result
What remains is:

\[ -2 + 3 = 1. \]

Thus, the value of the expression is

\[ \boxed{1}. \]