If ` a + (1)/(a) = - 2 ` then the value of `a ^(1000) + a ^(-1000) ` is

Options

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

Answer

To solve the equation \( a + \frac{1}{a} = -2 \) and find the value of \( a^{1000} + a^{-1000} \), we can follow these steps:

### Step 1: Solve for \( a \)
We start with the equation:
\[ a + \frac{1}{a} = -2 \]
To eliminate the fraction, we can multiply both sides by \( a \) (assuming \( a \neq 0 \)):
\[ a^2 + 1 = -2a \]
Rearranging this gives us a standard quadratic equation:
\[ a^2 + 2a + 1 = 0 \]

### Step 2: Factor the quadratic equation
The equation can be factored as:
\[ (a + 1)^2 = 0 \]
This implies:
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

### Step 3: Calculate \( a^{1000} + a^{-1000} \)
Now that we have \( a = -1 \), we can substitute this value into the expression we want to evaluate:
\[ a^{1000} + a^{-1000} = (-1)^{1000} + (-1)^{-1000} \]
Since \( (-1)^{1000} = 1 \) (because 1000 is an even number) and \( (-1)^{-1000} = \frac{1}{(-1)^{1000}} = \frac{1}{1} = 1 \), we have:
\[ a^{1000} + a^{-1000} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( a^{1000} + a^{-1000} \) is:
\[ \boxed{2} \]