If `x - (1)/(x) = 4`, then `x^(3) - (1)/(x^(3))` is equal to -

Options

A. 72
B. 68
C. 76
D. 64

Answer

To solve the equation \( x - \frac{1}{x} = 4 \) 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:
\[ x - \frac{1}{x} = 4 \]

### Step 2: Square both sides
To find \( x^2 + \frac{1}{x^2} \), we first square both sides of the equation:
\[ \left( x - \frac{1}{x} \right)^2 = 4^2 \]
This simplifies to:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 16 \]
\[ x^2 - 2 + \frac{1}{x^2} = 16 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 16 + 2 = 18 \]

### Step 4: Use the identity for cubes
Next, we use the identity for cubes:
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right) \left( x^2 + 1 + \frac{1}{x^2} \right) \]
We already know \( x - \frac{1}{x} = 4 \) and we found \( x^2 + \frac{1}{x^2} = 18 \).

### Step 5: Calculate \( x^3 - \frac{1}{x^3} \)
Now we can substitute these values into the identity:
\[ x^3 - \frac{1}{x^3} = 4 \left( 18 + 1 \right) = 4 \cdot 19 = 76 \]

### Conclusion
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{76} \]

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