The value of `3 + (1)/(sqrt(3)) + (1)/(3 + sqrt(3)) +(1)/(sqrt(3)-3)` is

Options

A. `3 + sqrt(3)`
B. `3`
C. 1
D. 0

Answer

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

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

### Step 2: Simplifying \(\frac{1}{3 + \sqrt{3}}\)
To simplify \(\frac{1}{3 + \sqrt{3}}\), we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}} = \frac{3 - \sqrt{3}}{(3 + \sqrt{3})(3 - \sqrt{3})} \]
Calculating the denominator:
\[ (3 + \sqrt{3})(3 - \sqrt{3}) = 9 - 3 = 6 \]
Thus,
\[ \frac{1}{3 + \sqrt{3}} = \frac{3 - \sqrt{3}}{6} \]

### Step 3: Simplifying \(\frac{1}{\sqrt{3} - 3}\)
Similarly, we simplify \(\frac{1}{\sqrt{3} - 3}\) by multiplying by the conjugate:
\[ \frac{1}{\sqrt{3} - 3} \cdot \frac{\sqrt{3} + 3}{\sqrt{3} + 3} = \frac{\sqrt{3} + 3}{(\sqrt{3} - 3)(\sqrt{3} + 3)} \]
Calculating the denominator:
\[ (\sqrt{3} - 3)(\sqrt{3} + 3) = 3 - 9 = -6 \]
Thus,
\[ \frac{1}{\sqrt{3} - 3} = \frac{\sqrt{3} + 3}{-6} = -\frac{\sqrt{3} + 3}{6} \]

### Step 4: Combining all parts
Now we substitute back into the expression:
\[ 3 + \frac{1}{\sqrt{3}} + \frac{3 - \sqrt{3}}{6} - \frac{\sqrt{3} + 3}{6} \]
Combining the last two fractions:
\[ \frac{3 - \sqrt{3} - \sqrt{3} - 3}{6} = \frac{-2\sqrt{3}}{6} = -\frac{\sqrt{3}}{3} \]
So the expression now looks like:
\[ 3 + \frac{1}{\sqrt{3}} - \frac{\sqrt{3}}{3} \]

### Step 5: Finding a common denominator
To combine \(\frac{1}{\sqrt{3}}\) and \(-\frac{\sqrt{3}}{3}\), we can express \(\frac{1}{\sqrt{3}}\) with a common denominator:
\[ \frac{1}{\sqrt{3}} = \frac{3}{3\sqrt{3}} \quad \text{and} \quad -\frac{\sqrt{3}}{3} = -\frac{\sqrt{3}}{3} \]
Now, we can combine:
\[ 3 + \left(\frac{3 - \sqrt{3}}{3\sqrt{3}}\right) \]

### Step 6: Final simplification
Now we can rewrite \(3\) as \(\frac{9\sqrt{3}}{3\sqrt{3}}\):
\[ \frac{9\sqrt{3}}{3\sqrt{3}} + \frac{3 - \sqrt{3}}{3\sqrt{3}} = \frac{9\sqrt{3} + 3 - \sqrt{3}}{3\sqrt{3}} = \frac{(9 - 1)\sqrt{3} + 3}{3\sqrt{3}} = \frac{8\sqrt{3} + 3}{3\sqrt{3}} \]

### Step 7: Conclusion
After simplifying, we find that the value of the entire expression is:
\[ 3 \]