If `x - (1)/(x) = 10`, then `x^(3) - (1)/(x^(3))` is equal to :

Options

A. 970
B. 1000
C. 1030
D. 1100

Answer

To solve the equation \( x - \frac{1}{x} = 10 \) and find \( x^3 - \frac{1}{x^3} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x - \frac{1}{x} = 10 \]

### Step 2: Cube both sides
To find \( x^3 - \frac{1}{x^3} \), we can use the identity:
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right)^3 + 3 \left( x - \frac{1}{x} \right) \]
Cubing both sides of the original equation gives:
\[ \left( x - \frac{1}{x} \right)^3 = 10^3 \]
Calculating \( 10^3 \):
\[ 10^3 = 1000 \]

### Step 3: Expand the left side using the identity
Using the identity mentioned:
\[ x^3 - \frac{1}{x^3} = 10^3 + 3 \left( x - \frac{1}{x} \right) \]
Substituting \( 10 \) for \( x - \frac{1}{x} \):
\[ x^3 - \frac{1}{x^3} = 1000 + 3 \times 10 \]

### Step 4: Calculate \( 3 \times 10 \)
Calculating \( 3 \times 10 \):
\[ 3 \times 10 = 30 \]

### Step 5: Add the results
Now, we add \( 1000 \) and \( 30 \):
\[ x^3 - \frac{1}{x^3} = 1000 + 30 = 1030 \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{1030} \]