The rule of `(1)/( 1 + sqrt2 + sqrt3) +(1)/( 1 - sqrt2 + sqrt3)` is

Options

A. 1
B. `4 ( sqrt3 + sqrt2)`
C. `sqrt3`
D. `sqrt2`

Answer

To solve the expression \( \frac{1}{1 + \sqrt{2} + \sqrt{3}} + \frac{1}{1 - \sqrt{2} + \sqrt{3}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{1}{1 + \sqrt{2} + \sqrt{3}} + \frac{1}{1 - \sqrt{2} + \sqrt{3}} \]

### Step 2: Find a common denominator
To combine the fractions, we need a common denominator. The common denominator will be:
\[ (1 + \sqrt{2} + \sqrt{3})(1 - \sqrt{2} + \sqrt{3}) \]

### Step 3: Expand the denominator
Using the difference of squares:
\[ (1 + \sqrt{3})^2 - (\sqrt{2})^2 = (1 + 2\sqrt{3} + 3) - 2 = 4 + 2\sqrt{3} \]

### Step 4: Cross multiply and add the numerators
Now we cross multiply:
\[ \frac{(1 - \sqrt{2} + \sqrt{3}) + (1 + \sqrt{2} + \sqrt{3})}{(1 + \sqrt{2} + \sqrt{3})(1 - \sqrt{2} + \sqrt{3})} \]

Simplifying the numerator:
\[ (1 - \sqrt{2} + \sqrt{3}) + (1 + \sqrt{2} + \sqrt{3}) = 2 + 2\sqrt{3} \]

### Step 5: Combine the results
Now we have:
\[ \frac{2 + 2\sqrt{3}}{4 + 2\sqrt{3}} \]

### Step 6: Simplify the expression
Factor out 2 from both the numerator and the denominator:
\[ \frac{2(1 + \sqrt{3})}{2(2 + \sqrt{3})} = \frac{1 + \sqrt{3}}{2 + \sqrt{3}} \]

### Step 7: Rationalize the denominator
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{(1 + \sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \]

Calculating the denominator:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1 \]

Calculating the numerator:
\[ (1 + \sqrt{3})(2 - \sqrt{3}) = 2 - \sqrt{3} + 2\sqrt{3} - 3 = -1 + \sqrt{3} \]

Thus, the expression simplifies to:
\[ \sqrt{3} - 1 \]

### Final Result
The final result of the expression is:
\[ \sqrt{3} - 1 \]