If `a ne 0 and a - (1)/(a)= 4`, find
`a^(4) + (1)/(a^(4))`

Answer

To solve the problem where \( a - \frac{1}{a} = 4 \) and we need to find \( a^4 + \frac{1}{a^4} \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ a - \frac{1}{a} = 4 \]
Now, we square both sides:
\[ \left(a - \frac{1}{a}\right)^2 = 4^2 \]

### Step 2: Expand the left side using the identity
Using the identity \( (x - y)^2 = x^2 - 2xy + y^2 \), we can expand the left side:
\[ a^2 - 2a\left(\frac{1}{a}\right) + \left(\frac{1}{a}\right)^2 = 16 \]
This simplifies to:
\[ a^2 - 2 + \frac{1}{a^2} = 16 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} - 2 = 16 \]
Adding 2 to both sides gives:
\[ a^2 + \frac{1}{a^2} = 16 + 2 = 18 \]

### Step 4: Square \( a^2 + \frac{1}{a^2} \) to find \( a^4 + \frac{1}{a^4} \)
Next, we square \( a^2 + \frac{1}{a^2} \):
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 18^2 \]
Expanding the left side using the identity \( (x + y)^2 = x^2 + 2xy + y^2 \):
\[ a^4 + 2 + \frac{1}{a^4} = 324 \]
Now, we can rearrange this to find \( a^4 + \frac{1}{a^4} \):
\[ a^4 + \frac{1}{a^4} = 324 - 2 = 322 \]

### Final Answer
Thus, the value of \( a^4 + \frac{1}{a^4} \) is:
\[ \boxed{322} \]