Find the value of: `1/(1+sqrt2)+1/(sqrt2+sqrt3)+1/(sqrt3+sqrt4)+...1/(sqrt99+sqrt100)`

Explanation

To solve the problem, we need to find the value of the expression:

\[ S = \frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \ldots + \frac{1}{\sqrt{99}+\sqrt{100}} \]

### Step 1: Rationalize Each Term

We start with the first term:

\[ \frac{1}{1+\sqrt{2}} \]

To rationalize the denominator, we multiply the numerator and denominator by \(1 - \sqrt{2}\):

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

### Step 2: Continue Rationalizing Subsequent Terms

Next, we rationalize the second term:

\[ \frac{1}{\sqrt{2}+\sqrt{3}} \]

Again, we multiply by \(\sqrt{2} - \sqrt{3}\):

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

### Step 3: Continue This Process

Continuing this process for the next terms, we find:

- For \(\frac{1}{\sqrt{3}+\sqrt{4}}\):

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

- For \(\frac{1}{\sqrt{4}+\sqrt{5}}\):

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

And so on, until we reach the last term:

\[ \frac{1}{\sqrt{99}+\sqrt{100}} = \sqrt{100} - \sqrt{99} = 10 - \sqrt{99} \]

### Step 4: Combine All Terms

Now, we can write the entire sum:

\[ S = (\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (2 - \sqrt{3}) + (\sqrt{4} - 2) + \ldots + (10 - \sqrt{99}) \]

### Step 5: Observe Cancellation

Notice that in this sum, all intermediate terms cancel out:

- \(-1\) from the first term remains.
- The \(\sqrt{2}\) from the first term cancels with the \(-\sqrt{2}\) from the second term.
- The \(\sqrt{3}\) cancels with the \(-\sqrt{3}\), and so on.

This leads to:

\[ S = 10 - 1 = 9 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{9} \]