Simplifiy `((3)/(2 +sqrt(3))-(2)/(2-sqrt(3)))/(2-5sqrt(3))`.

Options

A. 1
B. `-1`
C. `sqrt(3)`
D. `(sqrt(3))/(2)`

Answer

To simplify the expression \(\frac{\frac{3}{2 + \sqrt{3}} - \frac{2}{2 - \sqrt{3}}}{2 - 5\sqrt{3}}\), we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is \(\frac{3}{2 + \sqrt{3}} - \frac{2}{2 - \sqrt{3}}\). To combine these fractions, we need a common denominator.

**Hint:** The common denominator will be the product of the two denominators, which is \((2 + \sqrt{3})(2 - \sqrt{3})\).

### Step 2: Find the Common Denominator
The common denominator is:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1 \]
Thus, the common denominator is 1.

### Step 3: Rewrite Each Fraction
Now we can rewrite each fraction with the common denominator:
\[ \frac{3(2 - \sqrt{3}) - 2(2 + \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \]

### Step 4: Expand the Numerator
Expanding the numerator:
\[ 3(2 - \sqrt{3}) = 6 - 3\sqrt{3} \]
\[ 2(2 + \sqrt{3}) = 4 + 2\sqrt{3} \]
Thus, the numerator becomes:
\[ (6 - 3\sqrt{3}) - (4 + 2\sqrt{3}) = 6 - 3\sqrt{3} - 4 - 2\sqrt{3} = 2 - 5\sqrt{3} \]

### Step 5: Combine the Numerator
Now we have:
\[ \frac{2 - 5\sqrt{3}}{1} = 2 - 5\sqrt{3} \]

### Step 6: Divide by the Denominator
Now we divide this by the denominator \(2 - 5\sqrt{3}\):
\[ \frac{2 - 5\sqrt{3}}{2 - 5\sqrt{3}} = 1 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{1} \]