If x +`1/x` = `sqrt5`, then `x^(3) + 1/x^(3)` is equal to :

Options

A. `5sqrt(5)`
B. `2sqrt(5)`
C. `4sqrt(5)`
D. `3sqrt(5)`

Answer

To solve the problem, we need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x + \frac{1}{x} = \sqrt{5} \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + \frac{1}{x} = \sqrt{5} \]

2. **Cube both sides**:
\[ \left( x + \frac{1}{x} \right)^3 = \left( \sqrt{5} \right)^3 \]
This simplifies to:
\[ x^3 + 3x\left(\frac{1}{x}\right)\left(x + \frac{1}{x}\right) + \frac{1}{x^3} = 5\sqrt{5} \]

3. **Simplify the left-hand side**:
The term \( 3x\left(\frac{1}{x}\right) \) simplifies to \( 3 \):
\[ x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) = 5\sqrt{5} \]
Substitute \( x + \frac{1}{x} = \sqrt{5} \):
\[ x^3 + \frac{1}{x^3} + 3\sqrt{5} = 5\sqrt{5} \]

4. **Isolate \( x^3 + \frac{1}{x^3} \)**:
\[ x^3 + \frac{1}{x^3} = 5\sqrt{5} - 3\sqrt{5} \]
Simplifying gives:
\[ x^3 + \frac{1}{x^3} = 2\sqrt{5} \]

### Final Answer:
\[ x^3 + \frac{1}{x^3} = 2\sqrt{5} \]