If ` a-(1)/(a) = 3 , " then " a^6 + (1)/(a^6)` is equal to :

Options

A. 729
B. 996
C. 1331
D. 1298

Answer

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

### Step 1: Square the given equation
Start with the equation:
\[ a - \frac{1}{a} = 3 \]
Square both sides:
\[ \left(a - \frac{1}{a}\right)^2 = 3^2 \]
This simplifies to:
\[ a^2 - 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 9 \]
\[ a^2 - 2 + \frac{1}{a^2} = 9 \]
Adding 2 to both sides gives:
\[ a^2 + \frac{1}{a^2} = 11 \]

### Step 2: Find \( a^4 + \frac{1}{a^4} \)
Now, we will square \( a^2 + \frac{1}{a^2} \):
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 11^2 \]
This expands to:
\[ a^4 + 2 + \frac{1}{a^4} = 121 \]
Subtracting 2 from both sides gives:
\[ a^4 + \frac{1}{a^4} = 119 \]

### Step 3: Find \( a^6 + \frac{1}{a^6} \)
Next, we can use the identity:
\[ a^6 + \frac{1}{a^6} = (a^4 + \frac{1}{a^4})(a^2 + \frac{1}{a^2}) - (a^2 + \frac{1}{a^2}) \]
Substituting the values we found:
\[ a^6 + \frac{1}{a^6} = (119)(11) - 11 \]
Calculating \( 119 \times 11 \):
\[ 119 \times 11 = 1309 \]
Now, subtract 11:
\[ a^6 + \frac{1}{a^6} = 1309 - 11 = 1298 \]

### Final Answer
Thus, the value of \( a^6 + \frac{1}{a^6} \) is:
\[ \boxed{1298} \]