If `(x+1/x)=2` then `(x^3+1/x^3)` is equal to:

Options

A. 2
B. 4
C. 5
D. 6

Answer

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

1. **Given:** \( x + \frac{1}{x} = 2 \)

2. **Step 1:** Cube both sides of the equation.
\[ \left( x + \frac{1}{x} \right)^3 = 2^3 \]
This simplifies to:
\[ x^3 + 3\left( x \cdot \frac{1}{x} \right)\left( x + \frac{1}{x} \right) + \frac{1}{x^3} = 8 \]

3. **Step 2:** Simplify the expression.
Since \( x \cdot \frac{1}{x} = 1 \), we can substitute this into the equation:
\[ x^3 + 3(1)(x + \frac{1}{x}) + \frac{1}{x^3} = 8 \]
Now substitute \( x + \frac{1}{x} = 2 \):
\[ x^3 + 3(2) + \frac{1}{x^3} = 8 \]
This simplifies to:
\[ x^3 + 6 + \frac{1}{x^3} = 8 \]

4. **Step 3:** Isolate \( x^3 + \frac{1}{x^3} \).
\[ x^3 + \frac{1}{x^3} = 8 - 6 \]
Thus:
\[ x^3 + \frac{1}{x^3} = 2 \]

So, the final answer is:
\[ \boxed{2} \]