If ` a^(2) + a + 1 = 0` then the value of ` a^(9)` is

Options

A. 2
B. 3
C. 1
D. 0

Answer

To solve the equation \( a^2 + a + 1 = 0 \) and find the value of \( a^9 \), we can follow these steps:

### Step 1: Solve for \( a^2 \)
We start with the equation:
\[ a^2 + a + 1 = 0 \]
From this, we can express \( a^2 \) in terms of \( a \):
\[ a^2 = -a - 1 \]

### Step 2: Find \( a^3 \)
Next, we can find \( a^3 \) by multiplying both sides of the equation \( a^2 = -a - 1 \) by \( a \):
\[ a^3 = a \cdot a^2 = a(-a - 1) = -a^2 - a \]
Now, we substitute \( a^2 \) from Step 1:
\[ a^3 = -(-a - 1) - a = a + 1 - a = 1 \]

### Step 3: Calculate \( a^6 \)
Now that we have \( a^3 = 1 \), we can find \( a^6 \) by squaring \( a^3 \):
\[ a^6 = (a^3)^2 = 1^2 = 1 \]

### Step 4: Calculate \( a^9 \)
Finally, we can find \( a^9 \) by multiplying \( a^6 \) and \( a^3 \):
\[ a^9 = a^6 \cdot a^3 = 1 \cdot 1 = 1 \]

### Conclusion
Thus, the value of \( a^9 \) is:
\[ \boxed{1} \]