If `x+ (1)/(x)= -1` find
`x^(12) + (1)/(x^(12)) =` ?

Answer

To solve the equation \( x + \frac{1}{x} = -1 \) and find \( x^{12} + \frac{1}{x^{12}} \), we can follow these steps:

### Step 1: Cube both sides
Starting from the equation:
\[ x + \frac{1}{x} = -1 \]
We cube both sides:
\[ \left( x + \frac{1}{x} \right)^3 = (-1)^3 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right) = -1 \]

### Step 2: Substitute the known value
We know that \( x + \frac{1}{x} = -1 \). Substituting this into the equation gives:
\[ x^3 + \frac{1}{x^3} + 3(-1) = -1 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} - 3 = -1 \]
Adding 3 to both sides:
\[ x^3 + \frac{1}{x^3} = 2 \]

### Step 3: Find \( x^6 + \frac{1}{x^6} \)
Now we will use the identity for \( x^3 + \frac{1}{x^3} \) to find \( x^6 + \frac{1}{x^6} \):
\[ \left( x^3 + \frac{1}{x^3} \right)^2 = x^6 + \frac{1}{x^6} + 2 \]
Substituting \( x^3 + \frac{1}{x^3} = 2 \):
\[ (2)^2 = x^6 + \frac{1}{x^6} + 2 \]
This simplifies to:
\[ 4 = x^6 + \frac{1}{x^6} + 2 \]
Subtracting 2 from both sides:
\[ x^6 + \frac{1}{x^6} = 2 \]

### Step 4: Find \( x^{12} + \frac{1}{x^{12}} \)
Now we will use the identity for \( x^6 + \frac{1}{x^6} \) to find \( x^{12} + \frac{1}{x^{12}} \):
\[ \left( x^6 + \frac{1}{x^6} \right)^2 = x^{12} + \frac{1}{x^{12}} + 2 \]
Substituting \( x^6 + \frac{1}{x^6} = 2 \):
\[ (2)^2 = x^{12} + \frac{1}{x^{12}} + 2 \]
This simplifies to:
\[ 4 = x^{12} + \frac{1}{x^{12}} + 2 \]
Subtracting 2 from both sides:
\[ x^{12} + \frac{1}{x^{12}} = 2 \]

### Final Answer
Thus, the value of \( x^{12} + \frac{1}{x^{12}} \) is:
\[ \boxed{2} \]