`a + (1)/(a) + 2 =0` then the value of `a ^(37) - (1)/( a ^(100))` is

Options

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

Answer

To solve the equation \( a + \frac{1}{a} + 2 = 0 \) and find the value of \( a^{37} - \frac{1}{a^{100}} \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the equation:
\[ a + \frac{1}{a} + 2 = 0 \]
Rearranging gives:
\[ a + \frac{1}{a} = -2 \]

### Step 2: Multiplying by \( a \)
Multiply both sides by \( a \) (assuming \( a \neq 0 \)):
\[ a^2 + 1 = -2a \]
This can be rearranged to:
\[ a^2 + 2a + 1 = 0 \]

### Step 3: Factoring the Quadratic
The quadratic can be factored as:
\[ (a + 1)^2 = 0 \]
Thus, we find:
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

### Step 4: Finding \( a^{37} \) and \( \frac{1}{a^{100}} \)
Now, we need to calculate \( a^{37} \) and \( \frac{1}{a^{100}} \):
\[ a^{37} = (-1)^{37} = -1 \]
\[ \frac{1}{a^{100}} = \frac{1}{(-1)^{100}} = \frac{1}{1} = 1 \]

### Step 5: Calculating the Final Expression
Now substitute these values into the expression:
\[ a^{37} - \frac{1}{a^{100}} = -1 - 1 = -2 \]

### Conclusion
Thus, the value of \( a^{37} - \frac{1}{a^{100}} \) is:
\[ \boxed{-2} \]