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: Cube both sides of the equation
We start with the equation:
\[ x - \frac{1}{x} = 3 \]
Cubing both sides gives us:
\[ \left( x - \frac{1}{x} \right)^3 = 3^3 \]
This simplifies to:
\[ \left( x - \frac{1}{x} \right)^3 = 27 \]

### Step 2: Expand the left-hand side
Using the identity \( (a - b)^3 = a^3 - b^3 - 3ab(a - b) \), we can expand the left-hand side:
\[ x^3 - \left( \frac{1}{x} \right)^3 - 3 \left( x \cdot \frac{1}{x} \right) \left( x - \frac{1}{x} \right) = 27 \]
Since \( x \cdot \frac{1}{x} = 1 \), the equation becomes:
\[ x^3 - \frac{1}{x^3} - 3 \left( x - \frac{1}{x} \right) = 27 \]

### Step 3: Substitute the known value
We know from the original equation that \( x - \frac{1}{x} = 3 \). Substituting this into the equation gives:
\[ x^3 - \frac{1}{x^3} - 3(3) = 27 \]
This simplifies to:
\[ x^3 - \frac{1}{x^3} - 9 = 27 \]

### Step 4: Solve for \( x^3 - \frac{1}{x^3} \)
Now, we can isolate \( x^3 - \frac{1}{x^3} \):
\[ x^3 - \frac{1}{x^3} = 27 + 9 \]
Thus:
\[ x^3 - \frac{1}{x^3} = 36 \]

### Final Answer
The value of \( x^3 - \frac{1}{x^3} \) is \( 36 \).

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