If `(a + (1)/(a) ) ^(2) = 3` then the value of `a ^(6) - (1)/( a ^(6))` will be :

Options

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

Answer

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

\[ \left(a + \frac{1}{a}\right)^2 = 3 \]

### Step 1: Expand the left-hand side
Using the identity \((x + y)^2 = x^2 + y^2 + 2xy\), we can expand the left-hand side:

\[ a^2 + 2\left(a \cdot \frac{1}{a}\right) + \frac{1}{a^2} = 3 \]

This simplifies to:

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

### Step 2: Rearrange the equation
Now, we can rearrange this equation to isolate \(a^2 + \frac{1}{a^2}\):

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

### Step 3: Find \(a^4 + \frac{1}{a^4}\)
Next, we need to find \(a^4 + \frac{1}{a^4}\). We can use the identity:

\[ \left(a^2 + \frac{1}{a^2}\right)^2 = a^4 + 2 + \frac{1}{a^4} \]

Substituting \(a^2 + \frac{1}{a^2} = 1\):

\[ 1^2 = a^4 + 2 + \frac{1}{a^4} \]

This simplifies to:

\[ 1 = a^4 + 2 + \frac{1}{a^4} \]

Rearranging gives:

\[ a^4 + \frac{1}{a^4} = 1 - 2 = -1 \]

### Step 4: Find \(a^6 - \frac{1}{a^6}\)
Now we will find \(a^6 - \frac{1}{a^6}\). We can use the identity:

\[ a^6 - \frac{1}{a^6} = \left(a^2 + \frac{1}{a^2}\right)\left(a^4 - \frac{1}{a^4}\right) \]

We already have \(a^2 + \frac{1}{a^2} = 1\) and we found \(a^4 + \frac{1}{a^4} = -1\). Thus, we can find \(a^4 - \frac{1}{a^4}\):

\[ a^4 - \frac{1}{a^4} = (a^4 + \frac{1}{a^4}) - 2\left(\frac{1}{a^4}\right) = -1 - 2 = -3 \]

### Step 5: Calculate \(a^6 - \frac{1}{a^6}\)
Now substituting back into our equation:

\[ a^6 - \frac{1}{a^6} = \left(a^2 + \frac{1}{a^2}\right)\left(a^4 - \frac{1}{a^4}\right) = 1 \cdot (-3) = -3 \]

### Final Answer
Thus, the value of \(a^6 - \frac{1}{a^6}\) is:

\[ \boxed{-3} \]