If `x+(1)/(x)=13`, then the value of `x^(2)+(1)/(x^(2))` is:

Options

A. 165
B. 171
C. 167
D. 169

Answer

To solve the equation \( x + \frac{1}{x} = 13 \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x + \frac{1}{x} = 13 \]

### Step 2: Square both sides
To eliminate the fraction, we square both sides of the equation:
\[ \left(x + \frac{1}{x}\right)^2 = 13^2 \]

### Step 3: Expand the left side
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand the left side:
\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 169 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 169 \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} + 2 = 169 \]
Subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} = 169 - 2 \]

### Step 5: Calculate the final value
Now, we calculate:
\[ x^2 + \frac{1}{x^2} = 167 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{167} \]