If `(x+1/x)=3` the 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 need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x + \frac{1}{x} = 3 \).

### Step-by-Step Solution:

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

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

3. **Use the identity for the cube of a sum:**
The formula for \( (a + b)^3 \) is:
\[ a^3 + b^3 + 3ab(a + b) \]
Here, \( a = x \) and \( b = \frac{1}{x} \). Therefore, we have:
\[ x^3 + \frac{1}{x^3} + 3 \left( x \cdot \frac{1}{x} \right) \left( x + \frac{1}{x} \right) = 27 \]

4. **Simplify the expression:**
Since \( x \cdot \frac{1}{x} = 1 \), we can rewrite the equation as:
\[ x^3 + \frac{1}{x^3} + 3(1)(3) = 27 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 9 = 27 \]

5. **Isolate \( x^3 + \frac{1}{x^3} \):**
Now, subtract 9 from both sides:
\[ x^3 + \frac{1}{x^3} = 27 - 9 \]
Thus:
\[ x^3 + \frac{1}{x^3} = 18 \]

### Final Answer:
The value of \( x^3 + \frac{1}{x^3} \) is \( 18 \).

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