If ` x = 4 + sqrt(15)` then what is the value of ` [ x^(2)+((1)/( x^(2)))]` ?

Options

A. 62
B. 64
C. 34
D. 36

Answer

To find the value of \( x^2 + \frac{1}{x^2} \) when \( x = 4 + \sqrt{15} \), we can follow these steps:

### Step 1: Calculate \( \frac{1}{x} \)
To find \( \frac{1}{x} \), we start with:
\[ x = 4 + \sqrt{15} \]
To rationalize \( \frac{1}{x} \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1}{4 + \sqrt{15}} \cdot \frac{4 - \sqrt{15}}{4 - \sqrt{15}} = \frac{4 - \sqrt{15}}{(4 + \sqrt{15})(4 - \sqrt{15})} \]
Calculating the denominator:
\[ (4 + \sqrt{15})(4 - \sqrt{15}) = 4^2 - (\sqrt{15})^2 = 16 - 15 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 4 - \sqrt{15} \]

### Step 2: Calculate \( x + \frac{1}{x} \)
Now, we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (4 + \sqrt{15}) + (4 - \sqrt{15}) = 4 + \sqrt{15} + 4 - \sqrt{15} = 8 \]

### Step 3: Use the identity to find \( x^2 + \frac{1}{x^2} \)
We use the identity:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]
Substituting our value from Step 2:
\[ x^2 + \frac{1}{x^2} = 8^2 - 2 = 64 - 2 = 62 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{62} \]