If `(x ^(2) + (1)/( x ^(2)) =2),` then what is the value of `x ^(6)` ?

Options

A. 6
B. 0
C. 1
D. 3

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 2 \) and find the value of \( x^6 \), we can follow these steps:

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

### Step 2: Recognize a perfect square
We can express \( x^2 + \frac{1}{x^2} \) in terms of \( x + \frac{1}{x} \). We know that:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Thus, we can rearrange this to find:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]

### Step 3: Set the equation equal to zero
From our original equation:
\[ \left( x + \frac{1}{x} \right)^2 - 2 = 2 \]
This simplifies to:
\[ \left( x + \frac{1}{x} \right)^2 = 4 \]

### Step 4: Take the square root
Taking the square root of both sides gives us:
\[ x + \frac{1}{x} = 2 \quad \text{or} \quad x + \frac{1}{x} = -2 \]

### Step 5: Solve for \( x \)
If \( x + \frac{1}{x} = 2 \):
\[ x + \frac{1}{x} - 2 = 0 \]
Multiplying through by \( x \) gives:
\[ x^2 - 2x + 1 = 0 \]
Factoring this gives:
\[ (x - 1)^2 = 0 \implies x = 1 \]

If \( x + \frac{1}{x} = -2 \):
\[ x + \frac{1}{x} + 2 = 0 \]
Multiplying through by \( x \) gives:
\[ x^2 + 2x + 1 = 0 \]
Factoring this gives:
\[ (x + 1)^2 = 0 \implies x = -1 \]

### Step 6: Calculate \( x^6 \)
Now we can calculate \( x^6 \) for both values of \( x \):
1. If \( x = 1 \):
\[ x^6 = 1^6 = 1 \]
2. If \( x = -1 \):
\[ x^6 = (-1)^6 = 1 \]

### Conclusion
In both cases, we find that:
\[ x^6 = 1 \]

Thus, the value of \( x^6 \) is \( \boxed{1} \).