If ` x = (1)/( 2 + sqrt(3)) , y = (1)/( 2 - sqrt(3))` , then the value of ` 8 xy ( x^(2) + y^(2))` is

Options

A. 196
B. 290
C. 112
D. 194

Answer

To solve the problem, we need to find the value of \( 8xy(x^2 + y^2) \) given the values of \( x \) and \( y \).

### Step 1: Rationalize \( x \) and \( y \)

Given:
\[ x = \frac{1}{2 + \sqrt{3}}, \quad y = \frac{1}{2 - \sqrt{3}} \]

**Rationalizing \( x \):**
Multiply the numerator and denominator of \( x \) by the conjugate of the denominator:
\[ x = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \]
Calculating the denominator:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus,
\[ x = 2 - \sqrt{3} \]

**Rationalizing \( y \):**
Similarly, for \( y \):
\[ y = \frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})} \]
Calculating the denominator:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus,
\[ y = 2 + \sqrt{3} \]

### Step 2: Calculate \( xy \)

Now, we can calculate \( xy \):
\[ xy = (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 3: Calculate \( x^2 \) and \( y^2 \)

**Calculating \( x^2 \):**
\[ x^2 = (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

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

### Step 4: Calculate \( x^2 + y^2 \)

Now, we can find \( x^2 + y^2 \):
\[ x^2 + y^2 = (7 - 4\sqrt{3}) + (7 + 4\sqrt{3}) = 7 + 7 = 14 \]

### Step 5: Calculate \( 8xy(x^2 + y^2) \)

Now we can substitute the values into the expression \( 8xy(x^2 + y^2) \):
\[ 8xy(x^2 + y^2) = 8 \cdot 1 \cdot 14 = 112 \]

### Final Answer:
The value of \( 8xy(x^2 + y^2) \) is \( 112 \).

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