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

Options

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

Answer

To solve the equation \( x + \frac{1}{x} = 10 \) and find \( x^3 + \frac{1}{x^3} \), we can use the formula for the cube of a sum.

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 10 \]

2. **Use the identity for cubes:**
The identity we will use is:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) \]

3. **Substitute the value of \( x + \frac{1}{x} \):**
Substitute \( 10 \) into the identity:
\[ x^3 + \frac{1}{x^3} = 10^3 - 3 \cdot 10 \]

4. **Calculate \( 10^3 \):**
\[ 10^3 = 1000 \]

5. **Calculate \( 3 \cdot 10 \):**
\[ 3 \cdot 10 = 30 \]

6. **Combine the results:**
\[ x^3 + \frac{1}{x^3} = 1000 - 30 = 970 \]

Thus, the value of \( x^3 + \frac{1}{x^3} \) is \( 970 \).

### Final Answer:
\[ x^3 + \frac{1}{x^3} = 970 \]