If ` x^(3) + (1)/( x^(3)) = 110` then find the value of ` x + (1)/( x)`

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the equation \( x^3 + \frac{1}{x^3} = 110 \) and find the value of \( x + \frac{1}{x} \), we can use the algebraic identity for cubes.

### Step-by-Step Solution:

1. **Use the Identity**:
We know that:
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]
In our case, let \( a = x \) and \( b = \frac{1}{x} \). Thus, we can rewrite:
\[ \left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x \cdot \frac{1}{x} \right) \left( x + \frac{1}{x} \right) \]
This simplifies to:
\[ \left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right) \]

2. **Substitute the Given Value**:
We know from the problem statement that:
\[ x^3 + \frac{1}{x^3} = 110 \]
Substituting this into the identity gives us:
\[ \left( x + \frac{1}{x} \right)^3 = 110 + 3 \left( x + \frac{1}{x} \right) \]

3. **Let \( y = x + \frac{1}{x} \)**:
We can let \( y = x + \frac{1}{x} \). Thus, we can rewrite the equation as:
\[ y^3 = 110 + 3y \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ y^3 - 3y - 110 = 0 \]

5. **Finding the Roots**:
We can try to find rational roots using the Rational Root Theorem. Testing \( y = 5 \):
\[ 5^3 - 3(5) - 110 = 125 - 15 - 110 = 0 \]
Therefore, \( y = 5 \) is a root.

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

### Final Answer:
The value of \( x + \frac{1}{x} \) is \( 5 \).