The value of `((2+sqrt(3))/(2-sqrt(3))-4sqrt(3))^(2)` is

Options

A. a) `36`
B. b) `36sqrt(3)`
C. c) `49`
D. d) `49+sqrt(3)`

Answer

To solve the expression \(\left(\frac{2+\sqrt{3}}{2-\sqrt{3}} - 4\sqrt{3}\right)^{2}\), we will follow these steps:

### Step 1: Rationalize the fraction
We start with the expression \(\frac{2+\sqrt{3}}{2-\sqrt{3}}\). To simplify this, we multiply the numerator and denominator by the conjugate of the denominator, which is \(2+\sqrt{3}\):

\[ \frac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} \]

### Step 2: Simplify the denominator
The denominator simplifies as follows:

\[ (2-\sqrt{3})(2+\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 3: Simplify the numerator
Now, we simplify the numerator:

\[ (2+\sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 4: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{2+\sqrt{3}}{2-\sqrt{3}} = 7 + 4\sqrt{3} \]

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

\[ \left(7 + 4\sqrt{3} - 4\sqrt{3}\right)^{2} \]

### Step 6: Simplify the expression
The \(4\sqrt{3}\) terms cancel out:

\[ 7 + 4\sqrt{3} - 4\sqrt{3} = 7 \]

### Step 7: Square the result
Now we square the result:

\[ 7^{2} = 49 \]

### Final Answer
Thus, the value of \(\left(\frac{2+\sqrt{3}}{2-\sqrt{3}} - 4\sqrt{3}\right)^{2}\) is \(49\).

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