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

Options

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

Answer

To solve the problem, we need to find the value of \( a^5 + \frac{1}{a^5} \) given that \( a + \frac{1}{a} = 2 \).

### Step-by-Step Solution:

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

2. **Square both sides**:
\[ \left(a + \frac{1}{a}\right)^2 = 2^2 \]
This expands to:
\[ a^2 + 2 + \frac{1}{a^2} = 4 \]
Rearranging gives:
\[ a^2 + \frac{1}{a^2} = 4 - 2 = 2 \]

3. **Now, cube both sides of the original equation**:
\[ \left(a + \frac{1}{a}\right)^3 = 2^3 \]
Expanding this gives:
\[ a^3 + 3\left(a + \frac{1}{a}\right) + \frac{1}{a^3} = 8 \]
Substituting \( a + \frac{1}{a} = 2 \):
\[ a^3 + \frac{1}{a^3} + 3 \cdot 2 = 8 \]
Simplifying:
\[ a^3 + \frac{1}{a^3} + 6 = 8 \]
Therefore:
\[ a^3 + \frac{1}{a^3} = 8 - 6 = 2 \]

4. **Now we need to find \( a^5 + \frac{1}{a^5} \)**. We can use the identity:
\[ a^5 + \frac{1}{a^5} = \left(a^3 + \frac{1}{a^3}\right)\left(a^2 + \frac{1}{a^2}\right) - \left(a + \frac{1}{a}\right) \]
Substituting the values we found:
\[ a^5 + \frac{1}{a^5} = (2)(2) - 2 \]
Simplifying gives:
\[ a^5 + \frac{1}{a^5} = 4 - 2 = 2 \]

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