If x ` + 1/x = 2` then the value of `x^12 - (1)/(x^12)` is .

Options

A. (a) `-4`
B. (b) 4
C. (c) 2
D. (d) 0

Explanation

To solve the problem where \( x + \frac{1}{x} = 2 \) and we need to find the value of \( x^{12} - \frac{1}{x^{12}} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x + \frac{1}{x} = 2 \]

### Step 2: Square both sides
To find \( x^2 + \frac{1}{x^2} \), we square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 2^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 4 \]

### Step 3: Rearrange to find \( x^2 + \frac{1}{x^2} \)
Subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} = 4 - 2 = 2 \]

### Step 4: Find \( x^4 + \frac{1}{x^4} \)
Next, we square \( x^2 + \frac{1}{x^2} \):
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 2^2 \]
This expands to:
\[ x^4 + 2 + \frac{1}{x^4} = 4 \]
Rearranging gives:
\[ x^4 + \frac{1}{x^4} = 4 - 2 = 2 \]

### Step 5: Find \( x^8 + \frac{1}{x^8} \)
Now, we square \( x^4 + \frac{1}{x^4} \):
\[ \left(x^4 + \frac{1}{x^4}\right)^2 = 2^2 \]
This expands to:
\[ x^8 + 2 + \frac{1}{x^8} = 4 \]
Rearranging gives:
\[ x^8 + \frac{1}{x^8} = 4 - 2 = 2 \]

### Step 6: Find \( x^{12} - \frac{1}{x^{12}} \)
To find \( x^{12} - \frac{1}{x^{12}} \), we can use the identity:
\[ x^{12} - \frac{1}{x^{12}} = (x^8 + \frac{1}{x^8})(x^4 - \frac{1}{x^4}) \]
We already know:
\[ x^8 + \frac{1}{x^8} = 2 \quad \text{and} \quad x^4 + \frac{1}{x^4} = 2 \]
To find \( x^4 - \frac{1}{x^4} \), we use:
\[ x^4 - \frac{1}{x^4} = \sqrt{(x^4 + \frac{1}{x^4})^2 - 4} = \sqrt{2^2 - 4} = \sqrt{0} = 0 \]

### Step 7: Substitute back to find \( x^{12} - \frac{1}{x^{12}} \)
Now we substitute:
\[ x^{12} - \frac{1}{x^{12}} = (x^8 + \frac{1}{x^8})(x^4 - \frac{1}{x^4}) = 2 \cdot 0 = 0 \]

### Final Answer
Thus, the value of \( x^{12} - \frac{1}{x^{12}} \) is:
\[ \boxed{0} \]