If `a=2+sqrt3`, then what is the value of `(a^(2)+a^(-2))`?

Options

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

Answer

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

### Step 1: Calculate \( a^2 \)
We start with the expression for \( a \):
\[ a = 2 + \sqrt{3} \]
Now, we calculate \( a^2 \):
\[ a^2 = (2 + \sqrt{3})^2 \]
Using the identity \( (A + B)^2 = A^2 + 2AB + B^2 \):
\[ a^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
Calculating each term:
\[ = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 2: Calculate \( a^{-1} \)
Next, we find \( a^{-1} \):
\[ a^{-1} = \frac{1}{a} = \frac{1}{2 + \sqrt{3}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate \( 2 - \sqrt{3} \):
\[ a^{-1} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \]
Calculating the denominator:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, we have:
\[ a^{-1} = 2 - \sqrt{3} \]

### Step 3: Calculate \( a^{-2} \)
Now, we calculate \( a^{-2} \):
\[ a^{-2} = (a^{-1})^2 = (2 - \sqrt{3})^2 \]
Using the same identity:
\[ a^{-2} = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
Calculating each term:
\[ = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

### Step 4: Combine \( a^2 \) and \( a^{-2} \)
Now we can find \( a^2 + a^{-2} \):
\[ a^2 + a^{-2} = (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) \]
Combining the terms:
\[ = 7 + 4\sqrt{3} + 7 - 4\sqrt{3} = 14 \]

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