If `x - (1)/(x) = 6`, then `x^(3) - (1)/(x^(3))` is equal to :

Options

A. 216
B. 176
C. 234
D. 198

Answer

To solve the equation \( x - \frac{1}{x} = 6 \) and find the value of \( x^3 - \frac{1}{x^3} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x - \frac{1}{x} = 6 \]

### Step 2: Cube both sides
To find \( x^3 - \frac{1}{x^3} \), we can use the identity:
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right)^3 + 3 \left( x - \frac{1}{x} \right) \]
So, we will cube both sides:
\[ \left( x - \frac{1}{x} \right)^3 = 6^3 \]

### Step 3: Calculate \( 6^3 \)
Calculating \( 6^3 \):
\[ 6^3 = 216 \]

### Step 4: Substitute into the identity
Now we substitute back into the identity:
\[ x^3 - \frac{1}{x^3} = 216 + 3 \cdot 6 \]

### Step 5: Calculate \( 3 \cdot 6 \)
Calculating \( 3 \cdot 6 \):
\[ 3 \cdot 6 = 18 \]

### Step 6: Combine the results
Now, combine the results:
\[ x^3 - \frac{1}{x^3} = 216 + 18 = 234 \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{234} \]