If `(x^3+1/x^3)=2` then the value of `(x+1/x)` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( x + \frac{1}{x} \) given that \( x^3 + \frac{1}{x^3} = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x^3 + \frac{1}{x^3} = 2 \]

2. **Recall the identity**:
We can use the identity that relates \( x^3 + \frac{1}{x^3} \) to \( x + \frac{1}{x} \):
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) \]
Let \( y = x + \frac{1}{x} \). Then we can rewrite the equation as:
\[ y^3 - 3y = 2 \]

3. **Rearrange the equation**:
Rearranging gives us:
\[ y^3 - 3y - 2 = 0 \]

4. **Solve the cubic equation**:
We can try to find rational roots using the Rational Root Theorem. Testing \( y = 2 \):
\[ 2^3 - 3(2) - 2 = 8 - 6 - 2 = 0 \]
Thus, \( y = 2 \) is a root.

5. **Conclusion**:
Since \( y = x + \frac{1}{x} \), we have:
\[ x + \frac{1}{x} = 2 \]

### Final Answer:
The value of \( x + \frac{1}{x} \) is \( 2 \).

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