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

Options

A. `18sqrt3`
B. 18
C. `9sqrt3`
D. 12

Answer

To solve the problem, we start with the given equation:

1. **Given:**
\[ x + \frac{1}{x} = 3 \]

2. **Cube both sides:**
\[ \left(x + \frac{1}{x}\right)^3 = 3^3 \]
This simplifies to:
\[ \left(x + \frac{1}{x}\right)^3 = 27 \]

3. **Expand the left-hand side using the formula for the cube of a binomial:**
\[ a^3 + b^3 + 3ab(a + b) \]
Here, \(a = x\) and \(b = \frac{1}{x}\). Thus, we have:
\[ x^3 + \left(\frac{1}{x}\right)^3 + 3\left(x \cdot \frac{1}{x}\right)\left(x + \frac{1}{x}\right) \]
Since \(x \cdot \frac{1}{x} = 1\), this simplifies to:
\[ x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) \]

4. **Substituting the value of \(x + \frac{1}{x}\):**
\[ x^3 + \frac{1}{x^3} + 3 \cdot 3 = 27 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 9 = 27 \]

5. **Isolate \(x^3 + \frac{1}{x^3}\):**
\[ x^3 + \frac{1}{x^3} = 27 - 9 \]
Therefore:
\[ x^3 + \frac{1}{x^3} = 18 \]

So, the final answer is:
\[ \boxed{18} \]