If `a+1/a = -6`, find the value of `a^(3) + 1/a^(3)`

Options

A. 216
B. `-198`
C. `-216`
D. 198

Answer

To solve the problem where \( a + \frac{1}{a} = -6 \) and we need to find \( a^3 + \frac{1}{a^3} \), we can follow these steps:

### Step 1: Use the identity for \( a^3 + \frac{1}{a^3} \)

We know the formula:
\[ a^3 + \frac{1}{a^3} = \left( a + \frac{1}{a} \right)^3 - 3 \left( a + \frac{1}{a} \right) \]

### Step 2: Substitute the known value

From the problem, we have:
\[ a + \frac{1}{a} = -6 \]
Now, we can substitute this value into the formula:
\[ a^3 + \frac{1}{a^3} = (-6)^3 - 3(-6) \]

### Step 3: Calculate \( (-6)^3 \)

Calculating \( (-6)^3 \):
\[ (-6)^3 = -216 \]

### Step 4: Calculate \( -3(-6) \)

Calculating \( -3(-6) \):
\[ -3(-6) = 18 \]

### Step 5: Combine the results

Now, we combine the results from Steps 3 and 4:
\[ a^3 + \frac{1}{a^3} = -216 + 18 \]
\[ a^3 + \frac{1}{a^3} = -198 \]

### Final Answer

Thus, the value of \( a^3 + \frac{1}{a^3} \) is:
\[ \boxed{-198} \]