If `a + (1)/(a)= 6 and a ne 0`, find
`a^(2) - (1)/(a^(2))`

Answer

To solve the problem, we need to find \( a^2 - \frac{1}{a^2} \) given that \( a + \frac{1}{a} = 6 \) and \( a \neq 0 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ a + \frac{1}{a} = 6 \]

2. **Square both sides to find \( a^2 + 2 + \frac{1}{a^2} \):**
\[ \left(a + \frac{1}{a}\right)^2 = 6^2 \]
This expands to:
\[ a^2 + 2 + \frac{1}{a^2} = 36 \]

3. **Rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):**
\[ a^2 + \frac{1}{a^2} = 36 - 2 \]
Simplifying gives:
\[ a^2 + \frac{1}{a^2} = 34 \]

4. **Now, use the identity for \( a^2 - \frac{1}{a^2} \):**
We know that:
\[ a^2 - \frac{1}{a^2} = \left(a + \frac{1}{a}\right)\left(a - \frac{1}{a}\right) \]
To find \( a - \frac{1}{a} \), we can use the squared form:
\[ a - \frac{1}{a} = \sqrt{(a + \frac{1}{a})^2 - 4} = \sqrt{6^2 - 4} = \sqrt{36 - 4} = \sqrt{32} = 4\sqrt{2} \]

5. **Now substitute back to find \( a^2 - \frac{1}{a^2} \):**
\[ a^2 - \frac{1}{a^2} = (a + \frac{1}{a})(a - \frac{1}{a}) = 6 \times 4\sqrt{2} \]
Simplifying gives:
\[ a^2 - \frac{1}{a^2} = 24\sqrt{2} \]

### Final Answer:
\[ a^2 - \frac{1}{a^2} = 24\sqrt{2} \]