If `a ne 0 and a - (1)/(a)= 4`, find
`a^(2) + (1)/(a^(2))`

Answer

To solve the equation \( a - \frac{1}{a} = 4 \) and find \( a^2 + \frac{1}{a^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ a - \frac{1}{a} = 4 \]

### Step 2: Square both sides
To eliminate the fraction and find \( a^2 + \frac{1}{a^2} \), we square both sides of the equation:
\[ \left(a - \frac{1}{a}\right)^2 = 4^2 \]

### Step 3: Apply the square identity
Using the identity \( (x - y)^2 = x^2 - 2xy + y^2 \), we can expand the left side:
\[ a^2 - 2\left(a \cdot \frac{1}{a}\right) + \left(\frac{1}{a}\right)^2 = 16 \]
This simplifies to:
\[ a^2 - 2 + \frac{1}{a^2} = 16 \]

### Step 4: 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:
\[ a^2 + \frac{1}{a^2} = 16 + 2 \]

### Step 5: Final calculation
Thus, we find:
\[ a^2 + \frac{1}{a^2} = 18 \]

### Conclusion
The final answer is:
\[ \boxed{18} \]