If `x + 1/x = 3` , the value of `x^(5) + (1)/(x^5)` is

Options

A. 123
B. 126
C. 113
D. 129

Answer

To solve the problem, we need to find the value of \( x^5 + \frac{1}{x^5} \) given that \( x + \frac{1}{x} = 3 \).

### Step-by-step Solution:

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

2. **Square both sides:**
\[ \left( x + \frac{1}{x} \right)^2 = 3^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 9 \]

3. **Rearrange to find \( x^2 + \frac{1}{x^2} \):**
\[ x^2 + \frac{1}{x^2} = 9 - 2 = 7 \]

4. **Now, cube both sides of the original equation:**
\[ \left( x + \frac{1}{x} \right)^3 = 3^3 \]
This expands to:
\[ x^3 + 3\left( x + \frac{1}{x} \right) + \frac{1}{x^3} = 27 \]
Substituting \( x + \frac{1}{x} = 3 \):
\[ x^3 + \frac{1}{x^3} + 3 \cdot 3 = 27 \]
Simplifying gives:
\[ x^3 + \frac{1}{x^3} + 9 = 27 \]

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

6. **Now, we can find \( x^5 + \frac{1}{x^5} \) using the identity:**
\[ x^5 + \frac{1}{x^5} = \left( x^3 + \frac{1}{x^3} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we found:
\[ x^5 + \frac{1}{x^5} = (18)(7) - 3 \]

7. **Calculating the result:**
\[ x^5 + \frac{1}{x^5} = 126 - 3 = 123 \]

### Final Answer:
\[ x^5 + \frac{1}{x^5} = 123 \]

---