If `x+ (1)/(x)= -1` find
`x^(25) + (1)/(x^(25))=` ?

Answer

To solve the equation \( x + \frac{1}{x} = -1 \) and find \( x^{25} + \frac{1}{x^{25}} \), we can follow these steps:

### Step 1: Find \( x^2 + \frac{1}{x^2} \)

We start with the given equation:
\[ x + \frac{1}{x} = -1 \]
We square both sides:
\[ \left( x + \frac{1}{x} \right)^2 = (-1)^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 1 \]
Subtracting 2 from both sides gives:
\[ x^2 + \frac{1}{x^2} = 1 - 2 = -1 \]

### Step 2: Find \( x^3 + \frac{1}{x^3} \)

We can use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we found:
\[ x^3 + \frac{1}{x^3} = (-1)(-1) - (-1) \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} = 1 + 1 = 2 \]

### Step 3: Find \( x^4 + \frac{1}{x^4} \)

Using the identity:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Substituting the value we found:
\[ x^4 + \frac{1}{x^4} = (-1)^2 - 2 = 1 - 2 = -1 \]

### Step 4: Find \( x^5 + \frac{1}{x^5} \)

Using the identity:
\[ x^5 + \frac{1}{x^5} = \left( x + \frac{1}{x} \right) \left( x^4 + \frac{1}{x^4} \right) - \left( x^3 + \frac{1}{x^3} \right) \]
Substituting the values:
\[ x^5 + \frac{1}{x^5} = (-1)(-1) - 2 = 1 - 2 = -1 \]

### Step 5: Find \( x^{25} + \frac{1}{x^{25}} \)

We can continue this process, but we notice a pattern:
- \( x + \frac{1}{x} = -1 \)
- \( x^2 + \frac{1}{x^2} = -1 \)
- \( x^3 + \frac{1}{x^3} = 2 \)
- \( x^4 + \frac{1}{x^4} = -1 \)
- \( x^5 + \frac{1}{x^5} = -1 \)

From this, we can see that every 5th term seems to repeat the values. Specifically:
- \( x^1 + \frac{1}{x^1} = -1 \)
- \( x^2 + \frac{1}{x^2} = -1 \)
- \( x^3 + \frac{1}{x^3} = 2 \)
- \( x^4 + \frac{1}{x^4} = -1 \)
- \( x^5 + \frac{1}{x^5} = -1 \)

Since \( 25 \mod 5 = 0 \), we find:
\[ x^{25} + \frac{1}{x^{25}} = x^5 + \frac{1}{x^5} = -1 \]

### Final Answer
Thus, the final result is:
\[ \boxed{-1} \]