If `a + (1)/(a) = 3`, then the value of `(a^(6) + (1)/(a^(6)))` is equal to :

Options

A. 319
B. 322
C. 780
D. 730

Answer

To solve the problem \( a + \frac{1}{a} = 3 \) and find the value of \( a^6 + \frac{1}{a^6} \), we can follow these steps:

### Step 1: Find \( a^2 + \frac{1}{a^2} \)

We start with the equation:

\[ a + \frac{1}{a} = 3 \]

We can square both sides to find \( a^2 + \frac{1}{a^2} \):

\[ \left( a + \frac{1}{a} \right)^2 = a^2 + 2 + \frac{1}{a^2} \]

This simplifies to:

\[ 3^2 = a^2 + 2 + \frac{1}{a^2} \]

\[ 9 = a^2 + 2 + \frac{1}{a^2} \]

Subtracting 2 from both sides gives:

\[ a^2 + \frac{1}{a^2} = 9 - 2 = 7 \]

### Step 2: Find \( a^3 + \frac{1}{a^3} \)

Next, 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 known values:

\[ a^3 + \frac{1}{a^3} = 3 \cdot 7 - 3 \]

Calculating this gives:

\[ a^3 + \frac{1}{a^3} = 21 - 3 = 18 \]

### Step 3: Find \( a^6 + \frac{1}{a^6} \)

Now 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:

\[ a^6 + \frac{1}{a^6} = 18^2 - 2 \]

Calculating \( 18^2 \):

\[ 18^2 = 324 \]

Thus:

\[ a^6 + \frac{1}{a^6} = 324 - 2 = 322 \]

### Final Answer

Therefore, the value of \( a^6 + \frac{1}{a^6} \) is:

\[ \boxed{322} \]