If `x + (1)/(x) =-2,` then the value of `x ^(7) + (1)/(x^(7))`

Options

A. 0
B. 1
C. 2
D. `-2`

Answer

To solve the equation \( x + \frac{1}{x} = -2 \) and find the value of \( x^7 + \frac{1}{x^7} \), we can follow these steps:

### Step-by-Step Solution:

1. **Given Equation**:
We start with the equation:
\[ x + \frac{1}{x} = -2 \]

2. **Finding \( x \)**:
We can rearrange the equation to find \( x \). Let's multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = -2x \]
Rearranging gives us:
\[ x^2 + 2x + 1 = 0 \]
This can be factored as:
\[ (x + 1)^2 = 0 \]
Thus, we find:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

3. **Finding \( x^7 + \frac{1}{x^7} \)**:
Now that we have \( x = -1 \), we can calculate \( x^7 + \frac{1}{x^7} \):
\[ x^7 = (-1)^7 = -1 \]
\[ \frac{1}{x^7} = \frac{1}{(-1)^7} = -1 \]
Therefore:
\[ x^7 + \frac{1}{x^7} = -1 + (-1) = -2 \]

4. **Final Answer**:
The value of \( x^7 + \frac{1}{x^7} \) is:
\[ \boxed{-2} \]