What is the simplified value of `3+sqrt(3)+(1)/(3-sqrt(3))+(1)/(3+sqrt(3))` ?

Options

A. `2+sqrt(3)`
B. `2-sqrt(3)`
C. `4-sqrt(3)`
D. `4+sqrt(3)`

Answer

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

### Step 1: Combine the fractions
We need to simplify the two fractions \(\frac{1}{3 - \sqrt{3}}\) and \(\frac{1}{3 + \sqrt{3}}\). To do this, we will find a common denominator, which is \((3 - \sqrt{3})(3 + \sqrt{3})\).

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

### Step 2: Simplify the numerator
Now, we simplify the numerator:

\[ (3 + \sqrt{3}) + (3 - \sqrt{3}) = 3 + \sqrt{3} + 3 - \sqrt{3} = 6 \]

### Step 3: Simplify the denominator
Next, we simplify the denominator using the difference of squares:

\[ (3 - \sqrt{3})(3 + \sqrt{3}) = 3^2 - (\sqrt{3})^2 = 9 - 3 = 6 \]

### Step 4: Combine the results
Now, we can combine the results from steps 2 and 3:

\[ \frac{1}{3 - \sqrt{3}} + \frac{1}{3 + \sqrt{3}} = \frac{6}{6} = 1 \]

### Step 5: Substitute back into the original expression
Now, we substitute this result back into the original expression:

\[ 3 + \sqrt{3} + 1 \]

### Step 6: Final simplification
Finally, we simplify:

\[ 3 + 1 + \sqrt{3} = 4 + \sqrt{3} \]

Thus, the simplified value of the expression \(3 + \sqrt{3} + \frac{1}{3 - \sqrt{3}} + \frac{1}{3 + \sqrt{3}}\) is:

\[ \boxed{4 + \sqrt{3}} \]