`(1)/(sqrt(3)+sqrt(4))+(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 equal to

Options

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

Answer

To solve the expression

\[ \frac{1}{\sqrt{3}+\sqrt{4}} + \frac{1}{\sqrt{4}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{6}} + \frac{1}{\sqrt{6}+\sqrt{7}} + \frac{1}{\sqrt{7}+\sqrt{8}} + \frac{1}{\sqrt{8}+\sqrt{9}}, \]

we will rationalize each term in the series.

### Step 1: Rationalize the first term

For the first term,

\[ \frac{1}{\sqrt{3}+\sqrt{4}}, \]

we multiply the numerator and denominator by \(\sqrt{4}-\sqrt{3}\):

\[ \frac{\sqrt{4}-\sqrt{3}}{(\sqrt{3}+\sqrt{4})(\sqrt{4}-\sqrt{3})} = \frac{\sqrt{4}-\sqrt{3}}{4 - 3} = \sqrt{4} - \sqrt{3} = 2 - \sqrt{3}. \]

### Step 2: Rationalize the second term

For the second term,

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

we multiply the numerator and denominator by \(\sqrt{5}-\sqrt{4}\):

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

### Step 3: Rationalize the third term

For the third term,

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

we multiply the numerator and denominator by \(\sqrt{6}-\sqrt{5}\):

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

### Step 4: Rationalize the fourth term

For the fourth term,

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

we multiply the numerator and denominator by \(\sqrt{7}-\sqrt{6}\):

\[ \frac{\sqrt{7}-\sqrt{6}}{(\sqrt{6}+\sqrt{7})(\sqrt{7}-\sqrt{6})} = \frac{\sqrt{7}-\sqrt{6}}{7 - 6} = \sqrt{7} - \sqrt{6}. \]

### Step 5: Rationalize the fifth term

For the fifth term,

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

we multiply the numerator and denominator by \(\sqrt{8}-\sqrt{7}\):

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

### Step 6: Rationalize the sixth term

For the sixth term,

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

we multiply the numerator and denominator by \(\sqrt{9}-\sqrt{8}\):

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

### Step 7: Combine all the terms

Now we can combine all the rationalized terms:

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

Notice that all the intermediate terms cancel out:

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

### Final Result

Thus, the final result is:

\[ 3 - \sqrt{3}. \]