If `x+ (1)/(x)= -1` find
`x^(10) + (1)/(x^(10))`=?

Answer

To solve the equation \( x + \frac{1}{x} = -1 \) and find \( x^{10} + \frac{1}{x^{10}} \), we can follow these steps:

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

### Step 2: Simplify the equation
Using the identity \( a^3 + b^3 + 3ab(a + b) \):
\[ x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}) = -1 \]
Substituting \( x + \frac{1}{x} = -1 \):
\[ x^3 + \frac{1}{x^3} + 3(-1) = -1 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} - 3 = -1 \]
Thus:
\[ x^3 + \frac{1}{x^3} = -1 + 3 = 2 \]

### Step 3: Find \( x^6 + \frac{1}{x^6} \)
Now, we will find \( x^6 + \frac{1}{x^6} \) using the identity:
\[ x^6 + \frac{1}{x^6} = \left( x^3 + \frac{1}{x^3} \right)^2 - 2 \]
Substituting \( x^3 + \frac{1}{x^3} = 2 \):
\[ x^6 + \frac{1}{x^6} = 2^2 - 2 = 4 - 2 = 2 \]

### Step 4: Find \( x^{10} + \frac{1}{x^{10}} \)
Next, we will find \( x^{10} + \frac{1}{x^{10}} \) using the identity:
\[ x^{10} + \frac{1}{x^{10}} = (x^6 + \frac{1}{x^6})(x^4 + \frac{1}{x^4}) - (x^2 + \frac{1}{x^2}) \]
First, we need \( x^4 + \frac{1}{x^4} \):
\[ x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 \]
We find \( x^2 + \frac{1}{x^2} \) using:
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 = (-1)^2 - 2 = 1 - 2 = -1 \]
Now substituting back:
\[ x^4 + \frac{1}{x^4} = (-1)^2 - 2 = 1 - 2 = -1 \]

### Step 5: Substitute values into the equation for \( x^{10} + \frac{1}{x^{10}} \)
Now we can substitute:
\[ x^{10} + \frac{1}{x^{10}} = (2)(-1) - (-1) = -2 + 1 = -1 \]

### Final Answer
Thus, the value of \( x^{10} + \frac{1}{x^{10}} \) is:
\[ \boxed{-1} \]