Simplify
`1/(sqrt(100)-sqrt(99))-1/(sqrt(99)-sqrt(98))+1/(sqrt(98)-sqrt(97))-1/(sqrt(97)-sqrt(96))+........+1/(sqrt(2)-sqrt1)`

Options

A. 10
B. 9
C. 13
D. 11

Answer

To simplify the expression

\[ S = \frac{1}{\sqrt{100} - \sqrt{99}} - \frac{1}{\sqrt{99} - \sqrt{98}} + \frac{1}{\sqrt{98} - \sqrt{97}} - \frac{1}{\sqrt{97} - \sqrt{96}} + \ldots + \frac{1}{\sqrt{2} - \sqrt{1}} \]

we will rationalize each term in the series.

### Step 1: Rationalize Each Term
Let's start with the first term:

\[ \frac{1}{\sqrt{100} - \sqrt{99}} \]

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

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

This simplifies to:

\[ 10 + \sqrt{99} \]

### Step 2: Continue Rationalizing the Next Terms
Now, let's rationalize the second term:

\[ -\frac{1}{\sqrt{99} - \sqrt{98}} \cdot \frac{\sqrt{99} + \sqrt{98}}{\sqrt{99} + \sqrt{98}} = -\frac{\sqrt{99} + \sqrt{98}}{99 - 98} = -(\sqrt{99} + \sqrt{98}) \]

This simplifies to:

\[ -(\sqrt{99} + \sqrt{98}) \]

### Step 3: Combine the First Two Terms
Now we can combine the first two terms:

\[ (10 + \sqrt{99}) - (\sqrt{99} + \sqrt{98}) = 10 - \sqrt{98} \]

### Step 4: Continue with the Remaining Terms
Following this pattern, we can rationalize the next terms similarly:

\[ \frac{1}{\sqrt{98} - \sqrt{97}} = \sqrt{98} + \sqrt{97} \]

\[ -\frac{1}{\sqrt{97} - \sqrt{96}} = -(\sqrt{97} + \sqrt{96}) \]

Continuing this process, we will see that each positive term cancels with the negative term that follows it.

### Step 5: Recognize the Cancellation Pattern
Notice that each term cancels out with the subsequent term:

\[ S = (10 + \sqrt{99}) - (\sqrt{99} + \sqrt{98}) + (\sqrt{98} + \sqrt{97}) - (\sqrt{97} + \sqrt{96}) + \ldots - (\sqrt{2} - \sqrt{1}) \]

All terms cancel except for the very first term \(10\) and the last term \(-\sqrt{1}\).

### Final Step: Simplify the Remaining Expression
Thus, we have:

\[ S = 10 - 1 = 9 \]

### Conclusion
The simplified expression is:

\[ \boxed{9} \]