If `a + (1)/(a) = sqrt5,` then the value of `a ^(6) + (1)/(a ^(6))` is

Options

A. 18
B. 10
C. 12
D. 15

Answer

To solve the problem, we need to find the value of \( a^6 + \frac{1}{a^6} \) given that \( a + \frac{1}{a} = \sqrt{5} \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ a + \frac{1}{a} = \sqrt{5} \]

2. **Find \( a^2 + \frac{1}{a^2} \)**:
We can use the identity:
\[ a^2 + \frac{1}{a^2} = \left(a + \frac{1}{a}\right)^2 - 2 \]
Substituting the value we have:
\[ a^2 + \frac{1}{a^2} = \left(\sqrt{5}\right)^2 - 2 = 5 - 2 = 3 \]

3. **Find \( a^3 + \frac{1}{a^3} \)**:
We can use the identity:
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)\left(a^2 + \frac{1}{a^2}\right) - \left(a + \frac{1}{a}\right) \]
Substituting the values we have:
\[ a^3 + \frac{1}{a^3} = \sqrt{5} \cdot 3 - \sqrt{5} = 3\sqrt{5} - \sqrt{5} = 2\sqrt{5} \]

4. **Find \( a^6 + \frac{1}{a^6} \)**:
We can use the identity:
\[ a^6 + \frac{1}{a^6} = \left(a^3 + \frac{1}{a^3}\right)^2 - 2 \]
Substituting the value we found:
\[ a^6 + \frac{1}{a^6} = (2\sqrt{5})^2 - 2 = 4 \cdot 5 - 2 = 20 - 2 = 18 \]

Thus, the value of \( a^6 + \frac{1}{a^6} \) is \( \boxed{18} \).