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
We start with the equation:
\[ x + \frac{1}{x} = 1 \]
Now, we will cube both sides:
\[ \left(x + \frac{1}{x}\right)^3 = 1^3 \]
This gives us:
\[ x^3 + 3x\left(\frac{1}{x}\right)(x + \frac{1}{x}) + \frac{1}{x^3} = 1 \]

### Step 2: Simplify the Expression
Using the identity \( a^3 + b^3 + 3ab(a + b) \), we can rewrite the left side:
\[ x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}) = 1 \]
Substituting \( x + \frac{1}{x} = 1 \):
\[ x^3 + \frac{1}{x^3} + 3(1) = 1 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 3 = 1 \]

### Step 3: Solve for \( x^3 + \frac{1}{x^3} \)
Now, we can isolate \( x^3 + \frac{1}{x^3} \):
\[ x^3 + \frac{1}{x^3} = 1 - 3 \]
Thus:
\[ x^3 + \frac{1}{x^3} = -2 \]

### Step 4: Find \( x^{12} + \frac{1}{x^{12}} \)
Next, we need to find \( x^{12} + \frac{1}{x^{12}} \). We can use the identity:
\[ x^{n} + \frac{1}{x^{n}} = (x^{k} + \frac{1}{x^{k}})(x^{n-k} + \frac{1}{x^{n-k}}) - (x^{n-k-1} + \frac{1}{x^{n-k-1}}) \]
We will calculate \( x^6 + \frac{1}{x^6} \) first.

### Step 5: Calculate \( x^6 + \frac{1}{x^6} \)
Using the identity:
\[ x^6 + \frac{1}{x^6} = (x^3 + \frac{1}{x^3})^2 - 2 \]
Substituting \( x^3 + \frac{1}{x^3} = -2 \):
\[ x^6 + \frac{1}{x^6} = (-2)^2 - 2 = 4 - 2 = 2 \]

### Step 6: Calculate \( x^{12} + \frac{1}{x^{12}} \)
Now we can find \( x^{12} + \frac{1}{x^{12}} \):
\[ x^{12} + \frac{1}{x^{12}} = (x^6 + \frac{1}{x^6})^2 - 2 \]
Substituting \( x^6 + \frac{1}{x^6} = 2 \):
\[ x^{12} + \frac{1}{x^{12}} = (2)^2 - 2 = 4 - 2 = 2 \]

### Final Answer
Thus, the final result is:
\[ \boxed{2} \]