If `a + (1)/(a) = 2,` then the value of `a ^(5) + (1)/(a ^(5)) ` is

Options

A. 0
B. 1
C. 3
D. 2

Answer

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

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

2. **Find the value of \( a \):**
To find \( a \), we can rearrange the equation:
\[ a + \frac{1}{a} = 2 \]
Multiplying both sides by \( a \) (assuming \( a \neq 0 \)):
\[ a^2 + 1 = 2a \]
Rearranging gives us:
\[ a^2 - 2a + 1 = 0 \]
This can be factored as:
\[ (a - 1)^2 = 0 \]
Therefore, we find:
\[ a - 1 = 0 \implies a = 1 \]

3. **Calculate \( a^5 + \frac{1}{a^5} \):**
Now that we have \( a = 1 \), we can calculate \( a^5 + \frac{1}{a^5} \):
\[ a^5 + \frac{1}{a^5} = 1^5 + \frac{1}{1^5} = 1 + 1 = 2 \]

4. **Final Answer:**
Thus, the value of \( a^5 + \frac{1}{a^5} \) is:
\[ \boxed{2} \]