If $x - \frac{1}{x}$ = 4, then the value of $x^3 - \frac{1}{x^3}$ is:

Question

Accepted Answer

To solve the problem and justify why Option 2 is correct, we need to find the value of $ x^3 - \frac{1}{x^3} $ given that $ x - \frac{1}{x} = 4 $.

### Step-by-step Explanation:

1. **Given Equation:**
   $$
   x - \frac{1}{x} = 4
   $$

2. **Find $ x^2 + \frac{1}{x^2} $:**

We know the identity:
   $$
   \left(x - \frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2}
   $$

Substituting the given value:
   $$
   (x - \frac{1}{x})^2 = 4^2 = 16
   $$

Therefore:
   $$
   x^2 - 2 + \frac{1}{x^2} = 16
   $$

Solving for $ x^2 + \frac{1}{x^2} $:
   $$
   x^2 + \frac{1}{x^2} = 16 + 2 = 18
   $$

3. **Find $ x^3 - \frac{1}{x^3} $:**

We use the identity:
   $$
   x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)\left(x^2 + \frac{1}{x^2} + 1\right)
   $$

Substituting the known values:
   $$
   x^3 - \frac{1}{x^3} = 4 \times (18 + 1) = 4 \times 19 = 76
   $$

Thus, the value of $ x^3 - \frac{1}{x^3} $ is 76, which corresponds to Option 2.

### Explanation of Incorrect Options:

- **Option 1 (60):** This value does not satisfy the derived expression $ x^3 - \frac{1}{x^3} = 4 \times 19 $. The calculation clearly shows that the result should be 76, not 60.

- **Option 3 (64):** Similarly, 64 is not the result of the calculation $ 4 \times 19 $. The correct multiplication yields 76.

- **Option 4 (72):** This option is also incorrect because the multiplication of 4 and 19 results in 76, not 72.

Therefore, based on the calculations and logical reasoning, Option 2 is indeed the correct answer.