If x = 2 + `sqrt(3)` and y = 2 - `sqrt(3)`, find the value of `x^(-2) + y^(-2)`.

Options

A. 12
B. 14
C. 16
D. 18

Answer

To find the value of \( x^{-2} + y^{-2} \) where \( x = 2 + \sqrt{3} \) and \( y = 2 - \sqrt{3} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^{-2} + y^{-2} = \frac{1}{x^2} + \frac{1}{y^2} \]
This can be rewritten as:
\[ x^{-2} + y^{-2} = \frac{y^2 + x^2}{x^2 y^2} \]

### Step 2: Calculate \( x^2 \) and \( y^2 \)
First, we need to find \( x^2 \) and \( y^2 \):
\[ x^2 = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]
\[ y^2 = (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

### Step 3: Calculate \( x^2 + y^2 \)
Now, we add \( x^2 \) and \( y^2 \):
\[ x^2 + y^2 = (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) = 14 \]

### Step 4: Calculate \( xy \)
Next, we find \( xy \):
\[ xy = (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 5: Calculate \( x^2 y^2 \)
Now, we calculate \( x^2 y^2 \):
\[ x^2 y^2 = (xy)^2 = 1^2 = 1 \]

### Step 6: Substitute into the expression
Now we can substitute \( x^2 + y^2 \) and \( x^2 y^2 \) back into our expression:
\[ x^{-2} + y^{-2} = \frac{x^2 + y^2}{(xy)^2} = \frac{14}{1} = 14 \]

### Final Answer
Thus, the value of \( x^{-2} + y^{-2} \) is:
\[ \boxed{14} \]