If 'a' and 'b' are rational numbers and `(2+sqrt(3))/(2-sqrt(3)) = a+bsqrt(3) ` then `(a+b)^(2) =` ______ .

Options

A. 121
B. 171
C. 116
D. 198

Answer

To solve the equation \(\frac{2+\sqrt{3}}{2-\sqrt{3}} = a + b\sqrt{3}\), we will first simplify the left side of the equation.

### Step 1: Rationalize the denominator
To simplify \(\frac{2+\sqrt{3}}{2-\sqrt{3}}\), we multiply the numerator and the 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: Expand the numerator and denominator
Now we will expand both the numerator and the denominator:

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

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

### Step 3: Simplify the expression
Now, substituting back into the expression, we have:

\[ \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3} \]

### Step 4: Identify \(a\) and \(b\)
From the equation \(7 + 4\sqrt{3} = a + b\sqrt{3}\), we can identify:

- \(a = 7\)
- \(b = 4\)

### Step 5: Calculate \(a + b\)
Now we find \(a + b\):

\[ a + b = 7 + 4 = 11 \]

### Step 6: Calculate \((a + b)^2\)
Now we need to calculate \((a + b)^2\):

\[ (a + b)^2 = 11^2 = 121 \]

### Final Answer
Thus, \((a + b)^2 = 121\).

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