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 \( x^3 - \frac{1}{x^3} \), we can use a known algebraic identity.

### Step-by-Step Solution:

1. **Identify the given equation:**
\[ x - \frac{1}{x} = 7 \]
This is our starting point.

**Hint:** Recognize that we need to relate this expression to \( x^3 - \frac{1}{x^3} \).

2. **Use the identity for cubes:**
We know that:
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right)^3 + 3 \left( x - \frac{1}{x} \right) \]
Let \( k = x - \frac{1}{x} \). Thus, we can rewrite the expression as:
\[ x^3 - \frac{1}{x^3} = k^3 + 3k \]

**Hint:** Substitute \( k \) with the value from the given equation.

3. **Substitute the value of \( k \):**
Since \( k = 7 \):
\[ x^3 - \frac{1}{x^3} = 7^3 + 3 \cdot 7 \]

**Hint:** Calculate \( 7^3 \) and \( 3 \cdot 7 \) separately.

4. **Calculate \( 7^3 \):**
\[ 7^3 = 343 \]

**Hint:** Remember that \( 7 \times 7 = 49 \) and \( 49 \times 7 = 343 \).

5. **Calculate \( 3 \cdot 7 \):**
\[ 3 \cdot 7 = 21 \]

**Hint:** This is straightforward multiplication.

6. **Add the results:**
Now, combine the two results:
\[ x^3 - \frac{1}{x^3} = 343 + 21 = 364 \]

**Hint:** Ensure you add the numbers correctly.

7. **Final answer:**
Therefore, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{364} \]