If `a+(1)/(a)=-2`, then `a^(2)+(1)/(a^(2))` = ___________.

Answer

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

### Step-by-step Solution:

1. **Start with the given equation:**
\[ a + \frac{1}{a} = -2 \]

2. **Square both sides of the equation:**
\[ \left(a + \frac{1}{a}\right)^2 = (-2)^2 \]
This simplifies to:
\[ a^2 + 2\cdot a\cdot\frac{1}{a} + \frac{1}{a^2} = 4 \]
Since \( a\cdot\frac{1}{a} = 1 \), we have:
\[ a^2 + 2 + \frac{1}{a^2} = 4 \]

3. **Rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):**
\[ a^2 + \frac{1}{a^2} = 4 - 2 \]
This simplifies to:
\[ a^2 + \frac{1}{a^2} = 2 \]

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