If `x+ (1)/(x)= -2`, find
`x^(112) - (1)/(x^(113))=` ?

Answer

To solve the equation \( x + \frac{1}{x} = -2 \) and find \( x^{112} - \frac{1}{x^{113}} \), 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: Substitute \( x \) into the expression
Now we need to evaluate:
\[ x^{112} - \frac{1}{x^{113}} \]
Substituting \( x = -1 \):
\[ (-1)^{112} - \frac{1}{(-1)^{113}} \]

### Step 3: Calculate \( (-1)^{112} \) and \( (-1)^{113} \)
Since \( 112 \) is even:
\[ (-1)^{112} = 1 \]
And since \( 113 \) is odd:
\[ (-1)^{113} = -1 \implies \frac{1}{(-1)^{113}} = \frac{1}{-1} = -1 \]

### Step 4: Combine the results
Now substituting back into our expression:
\[ 1 - (-1) = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( x^{112} - \frac{1}{x^{113}} \) is:
\[ \boxed{2} \]