If `x - (1)/(x) = 7`, then `x^(3) - (1)/(x^(3))` is equal to -

Options

A. 480
B. 364
C. 376
D. 500

Answer

To solve the equation \( x - \frac{1}{x} = 7 \) 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:
\[ x - \frac{1}{x} = 7 \]

### 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:
\[ \left( x - \frac{1}{x} \right)^3 = 7^3 \]

### Step 3: Calculate \( 7^3 \)
Calculating \( 7^3 \):
\[ 7^3 = 343 \]

### Step 4: Substitute back into the identity
Now substitute back into the identity:
\[ x^3 - \frac{1}{x^3} = 343 + 3 \cdot 7 \]

### Step 5: Calculate \( 3 \cdot 7 \)
Calculating \( 3 \cdot 7 \):
\[ 3 \cdot 7 = 21 \]

### Step 6: Add the results
Now add \( 343 \) and \( 21 \):
\[ x^3 - \frac{1}{x^3} = 343 + 21 = 364 \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{364} \]