If ` x- (1)/ (x ) = 4`, find the value of :
` x ^(4) + ( 1 )/(x^(4))`

Answer

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

### Step 1: Square both sides of the equation
Starting with the equation:
\[ x - \frac{1}{x} = 4 \]
We square both sides:
\[ \left(x - \frac{1}{x}\right)^2 = 4^2 \]
This simplifies to:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 16 \]
which simplifies further to:
\[ x^2 - 2 + \frac{1}{x^2} = 16 \]

### Step 2: Rearrange the equation
Now, we rearrange the equation:
\[ x^2 + \frac{1}{x^2} - 2 = 16 \]
Adding 2 to both sides gives:
\[ x^2 + \frac{1}{x^2} = 18 \]

### Step 3: Use the result to find \( x^4 + \frac{1}{x^4} \)
We know that:
\[ x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 - 2 \]
Substituting \( x^2 + \frac{1}{x^2} = 18 \):
\[ x^4 + \frac{1}{x^4} = 18^2 - 2 \]

### Step 4: Calculate \( 18^2 \)
Calculating \( 18^2 \):
\[ 18^2 = 324 \]
Thus:
\[ x^4 + \frac{1}{x^4} = 324 - 2 = 322 \]

### Final Answer
The value of \( x^4 + \frac{1}{x^4} \) is:
\[ \boxed{322} \]