The value of
`1/(sqrt2+sqrt3)+1/(sqrt3+sqrt4)+1/(sqrt4+sqrt5)+1/(sqrt5+sqrt6)` will be

Options

A. `sqrt3(1+sqrt2)`
B. `sqrt2(sqrt3+1)`
C. `sqrt3(sqrt2-1)`
D. `sqrt2(sqrt3-1)`

Answer

To solve the expression \( \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \frac{1}{\sqrt{4}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{6}} \), we will rationalize each term in the sum.

### Step-by-Step Solution:

1. **First Term: Rationalize \( \frac{1}{\sqrt{2}+\sqrt{3}} \)**

Multiply the numerator and denominator by \( \sqrt{3} - \sqrt{2} \):
\[ \frac{1}{\sqrt{2}+\sqrt{3}} \cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{\sqrt{3}-\sqrt{2}}{(\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2})} \]
The denominator simplifies to:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Thus, the first term becomes:
\[ \sqrt{3} - \sqrt{2} \]

2. **Second Term: Rationalize \( \frac{1}{\sqrt{3}+\sqrt{4}} \)**

Multiply the numerator and denominator by \( \sqrt{4} - \sqrt{3} \):
\[ \frac{1}{\sqrt{3}+\sqrt{4}} \cdot \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}} = \frac{\sqrt{4}-\sqrt{3}}{(\sqrt{3}+\sqrt{4})(\sqrt{4}-\sqrt{3})} \]
The denominator simplifies to:
\[ (\sqrt{4})^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, the second term becomes:
\[ 2 - \sqrt{3} \]

3. **Third Term: Rationalize \( \frac{1}{\sqrt{4}+\sqrt{5}} \)**

Multiply the numerator and denominator by \( \sqrt{5} - \sqrt{4} \):
\[ \frac{1}{\sqrt{4}+\sqrt{5}} \cdot \frac{\sqrt{5}-\sqrt{4}}{\sqrt{5}-\sqrt{4}} = \frac{\sqrt{5}-\sqrt{4}}{(\sqrt{4}+\sqrt{5})(\sqrt{5}-\sqrt{4})} \]
The denominator simplifies to:
\[ (\sqrt{5})^2 - (\sqrt{4})^2 = 5 - 4 = 1 \]
Thus, the third term becomes:
\[ \sqrt{5} - 2 \]

4. **Fourth Term: Rationalize \( \frac{1}{\sqrt{5}+\sqrt{6}} \)**

Multiply the numerator and denominator by \( \sqrt{6} - \sqrt{5} \):
\[ \frac{1}{\sqrt{5}+\sqrt{6}} \cdot \frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}} = \frac{\sqrt{6}-\sqrt{5}}{(\sqrt{5}+\sqrt{6})(\sqrt{6}-\sqrt{5})} \]
The denominator simplifies to:
\[ (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]
Thus, the fourth term becomes:
\[ \sqrt{6} - \sqrt{5} \]

5. **Combine All Terms:**

Now, we can combine all the rationalized terms:
\[ (\sqrt{3} - \sqrt{2}) + (2 - \sqrt{3}) + (\sqrt{5} - 2) + (\sqrt{6} - \sqrt{5}) \]

Simplifying this:
- The \( \sqrt{3} \) terms cancel: \( \sqrt{3} - \sqrt{3} = 0 \)
- The \( 2 \) terms cancel: \( 2 - 2 = 0 \)
- The \( \sqrt{5} \) terms cancel: \( \sqrt{5} - \sqrt{5} = 0 \)

What remains is:
\[ -\sqrt{2} + \sqrt{6} \]

6. **Final Result:**

The final value is:
\[ \sqrt{6} - \sqrt{2} \]