If ` ( x^(4) + (1)/( x^(4))) = 198 and x gt 0` then what is the value of ` ( x^(2) - (1)/( x^(2)))`

Options

A. 14
B. `2 sqrt(7)`
C. `10 sqrt(7)`
D. 10

Answer

To solve the problem, we need to find the value of \( x^2 - \frac{1}{x^2} \) given that \( x^4 + \frac{1}{x^4} = 198 \) and \( x > 0 \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ x^4 + \frac{1}{x^4} = 198 \]

2. **Use the identity for \( x^4 + \frac{1}{x^4} \):**
We can express \( x^4 + \frac{1}{x^4} \) in terms of \( x^2 + \frac{1}{x^2} \):
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Let \( y = x^2 + \frac{1}{x^2} \). Then,
\[ x^4 + \frac{1}{x^4} = y^2 - 2 \]

3. **Substitute into the equation:**
Now we can substitute this into the original equation:
\[ y^2 - 2 = 198 \]

4. **Solve for \( y^2 \):**
Rearranging gives:
\[ y^2 = 198 + 2 = 200 \]

5. **Take the square root:**
Since \( x > 0 \), we take the positive root:
\[ y = \sqrt{200} = 10\sqrt{2} \]

6. **Relate \( y \) to \( x^2 - \frac{1}{x^2} \):**
We can now find \( x^2 - \frac{1}{x^2} \) using the identity:
\[ x^2 - \frac{1}{x^2} = \sqrt{(x^2 + \frac{1}{x^2})^2 - 4} \]
Substituting \( y \):
\[ x^2 - \frac{1}{x^2} = \sqrt{y^2 - 4} \]

7. **Calculate \( y^2 - 4 \):**
\[ y^2 - 4 = 200 - 4 = 196 \]

8. **Take the square root:**
\[ x^2 - \frac{1}{x^2} = \sqrt{196} = 14 \]

### Final Answer:
Thus, the value of \( x^2 - \frac{1}{x^2} \) is \( \boxed{14} \).