`1//(sqrt(3)-sqrt(2))` is not equal to

Options

A. `sqrt(3)+sqrt(2)`
B. `sqrt(2)//(sqrt(6)-2)`
C. `(sqrt(3)-sqrt(2))//(5-2sqrt(6))`
D. `sqrt(3)//(9-sqrt(6))`

Answer

To solve the problem \( \frac{1}{\sqrt{3} - \sqrt{2}} \) and determine which of the given options is not equal to this expression, we will rationalize the denominator and check each option step by step.

### Step-by-Step Solution:

1. **Rationalize the Denominator**:
We start with the expression \( \frac{1}{\sqrt{3} - \sqrt{2}} \). To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{3} + \sqrt{2} \).

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

2. **Calculate the Denominator**:
The denominator simplifies as follows:

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

Thus, we have:

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

3. **Evaluate Each Option**:
Now, we will check each option to see if they are equal to \( \sqrt{3} + \sqrt{2} \).

- **Option 1**: \( \sqrt{3} + \sqrt{2} \) (This is equal to our expression)
- **Option 2**: To check this, we multiply and divide by \( \sqrt{2} \):
\[ \frac{1}{\sqrt{3} - \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{6} - 2} \]
This is not equal to \( \sqrt{3} + \sqrt{2} \).
- **Option 3**: We multiply and divide by \( \sqrt{2} \):
\[ \frac{1}{\sqrt{3} - \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{6} - 2} \]
This simplifies to \( \sqrt{3} + \sqrt{2} \).
- **Option 4**: We multiply and divide by \( \sqrt{3} \):
\[ \frac{1}{\sqrt{3} - \sqrt{2}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3 - \sqrt{6}} \]
This does not equal \( \sqrt{3} + \sqrt{2} \).

4. **Conclusion**:
The option that is not equal to \( \frac{1}{\sqrt{3} - \sqrt{2}} \) is **Option 4**.

### Final Answer:
The expression \( \frac{1}{\sqrt{3} - \sqrt{2}} \) is not equal to **Option 4**.