If `x+ (1)/(x)= -2`, find
`x^(112) + (1)/(x^(112))=`?

Answer

To solve the equation \( x + \frac{1}{x} = -2 \) and find \( x^{112} + \frac{1}{x^{112}} \), we can follow these steps:

### Step 1: Solve for \( x \)
We start with the equation:
\[ x + \frac{1}{x} = -2 \]
To eliminate the fraction, we can multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = -2x \]
Rearranging gives us:
\[ x^2 + 2x + 1 = 0 \]
This can be factored as:
\[ (x + 1)^2 = 0 \]
Thus, we find:
\[ x + 1 = 0 \implies x = -1 \]

### Step 2: Calculate \( x^{112} + \frac{1}{x^{112}} \)
Now that we have \( x = -1 \), we can find \( x^{112} \):
\[ x^{112} = (-1)^{112} = 1 \]
Next, we calculate \( \frac{1}{x^{112}} \):
\[ \frac{1}{x^{112}} = \frac{1}{1} = 1 \]
Now we can add these two results:
\[ x^{112} + \frac{1}{x^{112}} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( x^{112} + \frac{1}{x^{112}} \) is:
\[ \boxed{2} \]