If `x - (1)/(x) = 3,` then what is the value of `x ^(3) - (1)/( x ^(3))` ?

Options

A. 36
B. 21
C. 9
D. 27

Answer

To solve the equation \( x - \frac{1}{x} = 3 \) 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:
\[ x - \frac{1}{x} = 3 \]

### Step 2: Square both sides
To find \( x^2 + \frac{1}{x^2} \), we square both sides of the equation:
\[ \left(x - \frac{1}{x}\right)^2 = 3^2 \]
This expands to:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 9 \]
Simplifying gives:
\[ x^2 - 2 + \frac{1}{x^2} = 9 \]

### Step 3: Rearrange to find \( x^2 + \frac{1}{x^2} \)
Now, we can rearrange the equation:
\[ x^2 + \frac{1}{x^2} = 9 + 2 = 11 \]

### Step 4: Use the identity for \( x^3 - \frac{1}{x^3} \)
We can 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 values we have:
\[ x^3 - \frac{1}{x^3} = (3)(11 + 1) \]
This simplifies to:
\[ x^3 - \frac{1}{x^3} = 3 \cdot 12 = 36 \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{36} \]