If `x-1/x=3`, then the value of `x^3-1/x^3` is

Options

A. 27
B. 32
C. 36
D. 42

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:
\[ x - \frac{1}{x} = 3 \]
Now, we will cube both sides:
\[ \left( x - \frac{1}{x} \right)^3 = 3^3 \]

### Step 2: Apply the identity for the cube of a binomial
Using the identity \( (a - b)^3 = a^3 - b^3 - 3ab(a - b) \), we can expand the left-hand side:
\[ \left( x - \frac{1}{x} \right)^3 = x^3 - \left( \frac{1}{x} \right)^3 - 3 \left( x \cdot \frac{1}{x} \right) \left( x - \frac{1}{x} \right) \]
This simplifies to:
\[ x^3 - \frac{1}{x^3} - 3 \cdot 1 \cdot \left( x - \frac{1}{x} \right) \]
Thus, we have:
\[ x^3 - \frac{1}{x^3} - 3 \left( x - \frac{1}{x} \right) = 27 \]

### Step 3: Substitute the known value
We know that \( x - \frac{1}{x} = 3 \). Substitute this value into the equation:
\[ x^3 - \frac{1}{x^3} - 3 \cdot 3 = 27 \]
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 \( x^3 - \frac{1}{x^3} \):
\[ x^3 - \frac{1}{x^3} = 27 + 9 \]
\[ x^3 - \frac{1}{x^3} = 36 \]

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