The value of `(1)/(1+sqrt(2))+(1)/(sqrt(2)+sqrt(3))+(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

Options

A. 2
B. 0
C. 4
D. 1

Answer

To solve the expression

\[ \frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \frac{1}{\sqrt{4}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{6}} + \frac{1}{\sqrt{6}+\sqrt{7}} + \frac{1}{\sqrt{7}+\sqrt{8}} + \frac{1}{\sqrt{8}+\sqrt{9}} \]

we will simplify each term using the technique of rationalizing the denominator.

### Step 1: Simplify the first term \(\frac{1}{1+\sqrt{2}}\)

Multiply the numerator and denominator by \(1 - \sqrt{2}\):

\[ \frac{1(1 - \sqrt{2})}{(1+\sqrt{2})(1-\sqrt{2})} = \frac{1 - \sqrt{2}}{1 - 2} = \frac{1 - \sqrt{2}}{-1} = \sqrt{2} - 1 \]

### Step 2: Simplify the second term \(\frac{1}{\sqrt{2}+\sqrt{3}}\)

Multiply the numerator and denominator by \(\sqrt{2} - \sqrt{3}\):

\[ \frac{1(\sqrt{2} - \sqrt{3})}{(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})} = \frac{\sqrt{2} - \sqrt{3}}{2 - 3} = \frac{\sqrt{2} - \sqrt{3}}{-1} = \sqrt{3} - \sqrt{2} \]

### Step 3: Simplify the third term \(\frac{1}{\sqrt{3}+\sqrt{4}}\)

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

\[ \frac{1(\sqrt{3} - 2)}{(\sqrt{3}+\sqrt{4})(\sqrt{3}-\sqrt{4})} = \frac{\sqrt{3} - 2}{3 - 4} = \frac{\sqrt{3} - 2}{-1} = 2 - \sqrt{3} \]

### Step 4: Simplify the fourth term \(\frac{1}{\sqrt{4}+\sqrt{5}}\)

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

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

### Step 5: Simplify the fifth term \(\frac{1}{\sqrt{5}+\sqrt{6}}\)

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

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

### Step 6: Simplify the sixth term \(\frac{1}{\sqrt{6}+\sqrt{7}}\)

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

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

### Step 7: Simplify the seventh term \(\frac{1}{\sqrt{7}+\sqrt{8}}\)

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

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

### Step 8: Simplify the eighth term \(\frac{1}{\sqrt{8}+\sqrt{9}}\)

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

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

### Step 9: Combine all the simplified terms

Now we can combine all the simplified terms:

\[ (\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (2 - \sqrt{3}) + (\sqrt{5} - 2) + (\sqrt{6} - \sqrt{5}) + (\sqrt{7} - \sqrt{6}) + (\sqrt{8} - \sqrt{7}) + (3 - \sqrt{8}) \]

### Step 10: Cancel out the terms

Notice that most terms cancel out:

- \(-1 + 2 - 2 + 3 = 2\)
- The square root terms cancel out: \(\sqrt{2} - \sqrt{2} + \sqrt{3} - \sqrt{3} + \sqrt{5} - \sqrt{5} + \sqrt{6} - \sqrt{6} + \sqrt{7} - \sqrt{7} + \sqrt{8} - \sqrt{8} = 0\)

Thus, the final value is:

\[ \boxed{2} \]