If `x + (1)/(x) = sqrt3,` then the value of `x ^(3) + (1)/(x ^(3))` is

Options

A. `sqrt3`
B. `(1)/(sqrt3`
C. 0
D. 1

Answer

To solve the problem where \( x + \frac{1}{x} = \sqrt{3} \) and we need to find the value of \( x^3 + \frac{1}{x^3} \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + \frac{1}{x} = \sqrt{3} \]

2. **Cube both sides**:
\[ \left( x + \frac{1}{x} \right)^3 = \left( \sqrt{3} \right)^3 \]

3. **Use the formula for the cube of a sum**:
The formula for \( (A + B)^3 \) is:
\[ A^3 + B^3 + 3AB(A + B) \]
Here, let \( A = x \) and \( B = \frac{1}{x} \). Thus,
\[ \left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x \cdot \frac{1}{x} \right) \left( x + \frac{1}{x} \right) \]

4. **Substituting values**:
Since \( x \cdot \frac{1}{x} = 1 \) and \( x + \frac{1}{x} = \sqrt{3} \), we can substitute these into the equation:
\[ \left( \sqrt{3} \right)^3 = x^3 + \frac{1}{x^3} + 3 \cdot 1 \cdot \sqrt{3} \]

5. **Calculate \( \left( \sqrt{3} \right)^3 \)**:
\[ \left( \sqrt{3} \right)^3 = 3\sqrt{3} \]

6. **Now, substitute back into the equation**:
\[ 3\sqrt{3} = x^3 + \frac{1}{x^3} + 3\sqrt{3} \]

7. **Isolate \( x^3 + \frac{1}{x^3} \)**:
Subtract \( 3\sqrt{3} \) from both sides:
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} \]
\[ x^3 + \frac{1}{x^3} = 0 \]

### Final Answer:
The value of \( x^3 + \frac{1}{x^3} \) is \( 0 \).