`(3+(1)/(sqrt(3))+(1)/(3+sqrt(3))+(1)/(sqrt(3)-3))` is equal to

Options

A. a) `1`
B. b) `3`
C. c) `3+sqrt(3)`
D. d) `3-sqrt(3)`

Answer

To solve the expression \( 3 + \frac{1}{\sqrt{3}} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3} \), we will proceed step by step.

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

### Step 2: Simplify the fractions
We will simplify the two fractions \( \frac{1}{3 + \sqrt{3}} \) and \( \frac{1}{\sqrt{3} - 3} \) by finding a common denominator.

#### For \( \frac{1}{3 + \sqrt{3}} \):
Multiply the numerator and denominator by \( 3 - \sqrt{3} \):
\[ \frac{1}{3 + \sqrt{3}} = \frac{3 - \sqrt{3}}{(3 + \sqrt{3})(3 - \sqrt{3})} = \frac{3 - \sqrt{3}}{9 - 3} = \frac{3 - \sqrt{3}}{6} \]

#### For \( \frac{1}{\sqrt{3} - 3} \):
Multiply the numerator and denominator by \( \sqrt{3} + 3 \):
\[ \frac{1}{\sqrt{3} - 3} = \frac{\sqrt{3} + 3}{(\sqrt{3} - 3)(\sqrt{3} + 3)} = \frac{\sqrt{3} + 3}{3 - 9} = \frac{\sqrt{3} + 3}{-6} = -\frac{\sqrt{3} + 3}{6} \]

### Step 3: Combine the fractions
Now we can combine the fractions:
\[ 3 + \frac{1}{\sqrt{3}} + \frac{3 - \sqrt{3}}{6} - \frac{\sqrt{3} + 3}{6} \]
Combine the last two fractions:
\[ \frac{3 - \sqrt{3} - \sqrt{3} - 3}{6} = \frac{-2\sqrt{3}}{6} = -\frac{\sqrt{3}}{3} \]

### Step 4: Rewrite the entire expression
Now we have:
\[ 3 + \frac{1}{\sqrt{3}} - \frac{\sqrt{3}}{3} \]

### Step 5: Simplify \( \frac{1}{\sqrt{3}} - \frac{\sqrt{3}}{3} \)
Convert \( \frac{1}{\sqrt{3}} \) to a common denominator:
\[ \frac{1}{\sqrt{3}} = \frac{3}{3\sqrt{3}} \]
Now combine:
\[ \frac{3}{3\sqrt{3}} - \frac{\sqrt{3}}{3} = \frac{3 - \sqrt{3} \cdot \sqrt{3}}{3\sqrt{3}} = \frac{3 - 3}{3\sqrt{3}} = \frac{0}{3\sqrt{3}} = 0 \]

### Step 6: Final Result
Thus, the entire expression simplifies to:
\[ 3 + 0 = 3 \]

### Conclusion
The value of the expression \( 3 + \frac{1}{\sqrt{3}} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3} \) is \( \boxed{3} \).