If ` x + (1)/( x) = 8 ` , then ` x^(2) + (1)/( x^(2))` is equal to :

Options

A. 62
B. 68
C. 64
D. 66

Answer

To solve the equation \( x + \frac{1}{x} = 8 \) and find \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x + \frac{1}{x} = 8 \]

### Step 2: Square both sides
To find \( x^2 + \frac{1}{x^2} \), we can square both sides of the equation:
\[ \left( x + \frac{1}{x} \right)^2 = 8^2 \]

### Step 3: Expand the left side
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we can expand the left side:
\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 64 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 64 \]

### Step 4: Isolate \( x^2 + \frac{1}{x^2} \)
Now, we can isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 64 - 2 \]
\[ x^2 + \frac{1}{x^2} = 62 \]

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

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