If ` x^(4) + (1)/( x^(4)) = 98 and x gt 1 ` then what is the value of `( x - (1)/( x))` ?

Options

A. 2
B. ` 2 sqrt(2)`
C. ` sqrt(5)`
D. ` sqrt(3)`

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 98 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^4 + \frac{1}{x^4} = 98 \]

### Step 2: Add 2 to both sides
We can add 2 to both sides of the equation:
\[ x^4 + \frac{1}{x^4} + 2 = 100 \]

### Step 3: Recognize the square
Notice that \( x^4 + 2 + \frac{1}{x^4} \) can be rewritten as:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 \]
Thus, we have:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 100 \]

### Step 4: Take the square root
Taking the square root of both sides gives us:
\[ x^2 + \frac{1}{x^2} = 10 \]

### Step 5: Add 2 again
Next, we add 2 to both sides:
\[ x^2 + \frac{1}{x^2} + 2 = 12 \]

### Step 6: Recognize the square again
This can be rewritten as:
\[ \left( x + \frac{1}{x} \right)^2 = 12 \]

### Step 7: Take the square root again
Taking the square root of both sides gives us:
\[ x + \frac{1}{x} = \sqrt{12} = 2\sqrt{3} \]

### Step 8: Use the identity for \( x - \frac{1}{x} \)
We can use the identity:
\[ \left( x - \frac{1}{x} \right)^2 = \left( x + \frac{1}{x} \right)^2 - 4 \]
Substituting the value we found:
\[ \left( x - \frac{1}{x} \right)^2 = (2\sqrt{3})^2 - 4 = 12 - 4 = 8 \]

### Step 9: Take the square root
Taking the square root gives us:
\[ x - \frac{1}{x} = \sqrt{8} = 2\sqrt{2} \]

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