Rationalize the denominator of : `(1)/(sqrt(3)-sqrt(2)+1)`

Answer

To rationalize the denominator of the expression \( \frac{1}{\sqrt{3} - \sqrt{2} + 1} \), we will follow these steps:

### Step 1: Rewrite the Denominator
We can rewrite the denominator as:
\[ \sqrt{3} + 1 - \sqrt{2} \]
This helps in visualizing the rationalization process.

### Step 2: Multiply by the Conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{3} + 1 - \sqrt{2} \) is \( \sqrt{3} + 1 + \sqrt{2} \). Thus, we multiply:
\[ \frac{1}{\sqrt{3} - \sqrt{2} + 1} \times \frac{\sqrt{3} + 1 + \sqrt{2}}{\sqrt{3} + 1 + \sqrt{2}} \]

### Step 3: Simplify the Denominator
Now, we simplify the denominator using the difference of squares:
\[ (\sqrt{3} + 1 - \sqrt{2})(\sqrt{3} + 1 + \sqrt{2}) = (\sqrt{3} + 1)^2 - (\sqrt{2})^2 \]
Calculating \( (\sqrt{3} + 1)^2 \):
\[ (\sqrt{3})^2 + 2 \cdot \sqrt{3} \cdot 1 + 1^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
Now, substituting back:
\[ (4 + 2\sqrt{3}) - 2 = 2 + 2\sqrt{3} \]

### Step 4: Simplify the Numerator
The numerator becomes:
\[ \sqrt{3} + 1 + \sqrt{2} \]
So, we have:
\[ \frac{\sqrt{3} + 1 + \sqrt{2}}{2 + 2\sqrt{3}} \]

### Step 5: Factor Out the Denominator
We can factor out a 2 from the denominator:
\[ \frac{\sqrt{3} + 1 + \sqrt{2}}{2(1 + \sqrt{3})} \]
This simplifies to:
\[ \frac{1}{2} \cdot \frac{\sqrt{3} + 1 + \sqrt{2}}{1 + \sqrt{3}} \]

### Step 6: Rationalize Again
To further rationalize, we multiply the numerator and denominator by \( 1 - \sqrt{3} \):
\[ \frac{1}{2} \cdot \frac{(\sqrt{3} + 1 + \sqrt{2})(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} \]

### Step 7: Simplify the Denominator Again
The denominator simplifies to:
\[ 1 - 3 = -2 \]

### Step 8: Expand the Numerator
Now we expand the numerator:
\[ (\sqrt{3} + 1 + \sqrt{2})(1 - \sqrt{3}) = \sqrt{3} - 3 + 1 - \sqrt{3} + \sqrt{2} - \sqrt{6} \]
Combining like terms:
\[ -2 + \sqrt{2} - \sqrt{6} \]

### Final Expression
Putting it all together, we have:
\[ \frac{1}{2} \cdot \frac{-2 + \sqrt{2} - \sqrt{6}}{-2} = \frac{1}{4}(\sqrt{2} - \sqrt{6} + 2) \]

Thus, the final answer is:
\[ \frac{\sqrt{2} - \sqrt{6} + 2}{4} \]