If `a- (1)/(a)= 8 and a ne 0`, find:
`a+ (1)/(a)`

Answer

To solve the problem, we need to find \( a + \frac{1}{a} \) given that \( a - \frac{1}{a} = 8 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ a - \frac{1}{a} = 8 \]

2. **Use the identity for squares:**
We know that:
\[ (a + \frac{1}{a})^2 - (a - \frac{1}{a})^2 = 4 \]
This is derived from the difference of squares formula \( x^2 - y^2 = (x+y)(x-y) \).

3. **Let \( x = a + \frac{1}{a} \) and \( y = a - \frac{1}{a} \):**
Thus, we can rewrite the equation as:
\[ x^2 - y^2 = 4 \]
Substituting \( y = 8 \):
\[ x^2 - 8^2 = 4 \]

4. **Calculate \( 8^2 \):**
\[ 8^2 = 64 \]
So, we have:
\[ x^2 - 64 = 4 \]

5. **Rearranging the equation:**
\[ x^2 = 64 + 4 \]
\[ x^2 = 68 \]

6. **Taking the square root:**
\[ x = \sqrt{68} \]
We can simplify \( \sqrt{68} \):
\[ \sqrt{68} = \sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17} \]

7. **Final result:**
Therefore, we find:
\[ a + \frac{1}{a} = 2\sqrt{17} \]

### Final Answer:
\[ a + \frac{1}{a} = 2\sqrt{17} \]