If `x-x^(-1)=6`, then the value of `x^(4)+x^(-4)` is:

Options

A. 1158
B. 1154
C. 1440
D. 1442

Answer

To solve the equation \( x - x^{-1} = 6 \) and find the value of \( x^4 + x^{-4} \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ x - x^{-1} = 6 \]
We square both sides:
\[ (x - x^{-1})^2 = 6^2 \]
This gives us:
\[ x^2 - 2 + x^{-2} = 36 \]

### Step 2: Rearrange the equation
Now, we rearrange the equation:
\[ x^2 + x^{-2} - 2 = 36 \]
Adding 2 to both sides, we have:
\[ x^2 + x^{-2} = 36 + 2 = 38 \]

### Step 3: Square \( x^2 + x^{-2} \)
Next, we need to find \( x^4 + x^{-4} \). We can do this by squaring \( x^2 + x^{-2} \):
\[ (x^2 + x^{-2})^2 = 38^2 \]
This expands to:
\[ x^4 + 2 + x^{-4} = 1444 \]

### Step 4: Rearrange to find \( x^4 + x^{-4} \)
Now, we rearrange to isolate \( x^4 + x^{-4} \):
\[ x^4 + x^{-4} = 1444 - 2 = 1442 \]

### Final Answer
Thus, the value of \( x^4 + x^{-4} \) is:
\[ \boxed{1442} \]