If `x+ (1)/(x)=1`, find
`x^(9) + (1)/(x^(9))=`

Answer

To solve the equation \( x + \frac{1}{x} = 1 \) and find \( x^9 + \frac{1}{x^9} \), we can follow these steps:

### Step 1: Cube both sides of the equation
We start with the equation:
\[ x + \frac{1}{x} = 1 \]
Cubing both sides gives us:
\[ \left( x + \frac{1}{x} \right)^3 = 1^3 \]
This expands to:
\[ x^3 + 3x \cdot \frac{1}{x} \cdot \left( x + \frac{1}{x} \right) + \frac{1}{x^3} = 1 \]

### Step 2: Simplify the equation
The term \( 3x \cdot \frac{1}{x} \) simplifies to \( 3 \), and since \( x + \frac{1}{x} = 1 \), we can substitute:
\[ x^3 + 3 \cdot 1 + \frac{1}{x^3} = 1 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 3 = 1 \]

### Step 3: Isolate \( x^3 + \frac{1}{x^3} \)
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 \( x^9 + \frac{1}{x^9} \)
To find \( x^9 + \frac{1}{x^9} \), we can use the identity:
\[ x^9 + \frac{1}{x^9} = \left( x^3 + \frac{1}{x^3} \right)^3 - 3 \left( x^3 + \frac{1}{x^3} \right) \]
Substituting \( x^3 + \frac{1}{x^3} = -2 \):
\[ x^9 + \frac{1}{x^9} = (-2)^3 - 3(-2) \]

### Step 5: Calculate the values
Calculating \( (-2)^3 \):
\[ (-2)^3 = -8 \]
And calculating \( -3(-2) \):
\[ -3(-2) = 6 \]
Now substituting back:
\[ x^9 + \frac{1}{x^9} = -8 + 6 = -2 \]

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