If `a+ (1)/(a)= -1` find `a^(3) + 3`= ?

Answer

To solve the equation \( a + \frac{1}{a} = -1 \) and find the value of \( a^3 + 3 \), we can follow these steps:

### Step 1: Cube both sides of the equation
We start with the equation:
\[ a + \frac{1}{a} = -1 \]
We will cube both sides:
\[ \left(a + \frac{1}{a}\right)^3 = (-1)^3 \]

### Step 2: Apply the cube expansion formula
Using the formula for the cube of a sum, \( (x + y)^3 = x^3 + y^3 + 3xy(x + y) \), we can expand the left side:
\[ a^3 + \left(\frac{1}{a}\right)^3 + 3 \cdot a \cdot \frac{1}{a} \left(a + \frac{1}{a}\right) = -1 \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} + 3 \cdot 1 \cdot (-1) = -1 \]
Thus, we have:
\[ a^3 + \frac{1}{a^3} - 3 = -1 \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate \( a^3 + \frac{1}{a^3} \):
\[ a^3 + \frac{1}{a^3} = -1 + 3 \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} = 2 \]

### Step 4: Find \( a^3 \)
We know from the previous step that:
\[ a^3 + \frac{1}{a^3} = 2 \]
Let \( x = a^3 \). Then we can write:
\[ x + \frac{1}{x} = 2 \]
Multiplying through by \( x \) gives:
\[ x^2 + 1 = 2x \]
Rearranging this leads to:
\[ x^2 - 2x + 1 = 0 \]
This factors to:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x = 1 \quad \text{(which means } a^3 = 1\text{)} \]

### Step 5: Calculate \( a^3 + 3 \)
Finally, we need to find \( a^3 + 3 \):
\[ a^3 + 3 = 1 + 3 = 4 \]

### Final Answer
Thus, the value of \( a^3 + 3 \) is:
\[ \boxed{4} \]