Simplify `((3/(2+sqrt3)-2/(2-sqrt3)))/(2-5sqrt3)`

Options

A. `1/2-5sqrt3`
B. 1
C. `2-5sqrt3`
D. 0

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: Find a common denominator for the fractions in the numerator.
The common denominator for \(2+\sqrt{3}\) and \(2-\sqrt{3}\) is \((2+\sqrt{3})(2-\sqrt{3})\).

### Step 2: Rewrite the fractions with the common denominator.
\[ \frac{3}{2+\sqrt{3}} = \frac{3(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} \]
\[ \frac{2}{2-\sqrt{3}} = \frac{2(2+\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} \]

### Step 3: Combine the fractions in the numerator.
Now, we can combine the fractions:
\[ \frac{3(2-\sqrt{3}) - 2(2+\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} \]

### Step 4: Simplify the numerator.
Expanding the numerator:
\[ 3(2-\sqrt{3}) = 6 - 3\sqrt{3} \]
\[ 2(2+\sqrt{3}) = 4 + 2\sqrt{3} \]
Now, substituting back into the numerator:
\[ 6 - 3\sqrt{3} - (4 + 2\sqrt{3}) = 6 - 3\sqrt{3} - 4 - 2\sqrt{3} = 2 - 5\sqrt{3} \]

### Step 5: Write the expression with the simplified numerator.
Now, our expression looks like this:
\[ \frac{2 - 5\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})} \]

### Step 6: Simplify the denominator.
The denominator can be simplified using the difference of squares:
\[ (2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 7: Final expression.
Thus, we have:
\[ \frac{2 - 5\sqrt{3}}{1} = 2 - 5\sqrt{3} \]

### Step 8: Divide by the denominator \(2 - 5\sqrt{3}\).
Now we divide by \(2 - 5\sqrt{3}\):
\[ \frac{2 - 5\sqrt{3}}{2 - 5\sqrt{3}} = 1 \]

### Final Answer:
The simplified expression is \(1\).

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