If `(a+(1)/(a))^(2)= 3`, then `a^(3)+(1)/(a^(3))` equals

Options

A. `6 sqrt(3)`
B. `3 sqrt(3)`
C. `0`
D. `7 sqrt(7)`

Explanation

To solve the equation \((a + \frac{1}{a})^2 = 3\) and find the value of \(a^3 + \frac{1}{a^3}\), we can follow these steps:

### Step 1: Start with the given equation
\[ (a + \frac{1}{a})^2 = 3 \]

### Step 2: Take the square root of both sides
Taking the square root gives us:
\[ a + \frac{1}{a} = \sqrt{3} \quad \text{(or } a + \frac{1}{a} = -\sqrt{3} \text{, but we will consider positive root for simplicity)} \]

### Step 3: Cube both sides
Now, we will cube both sides of the equation:
\[ (a + \frac{1}{a})^3 = (\sqrt{3})^3 \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} + 3(a)(\frac{1}{a})(a + \frac{1}{a}) = 3\sqrt{3} \]

### Step 4: Simplify the equation
Since \(a \cdot \frac{1}{a} = 1\), we can rewrite the equation as:
\[ a^3 + \frac{1}{a^3} + 3(a + \frac{1}{a}) = 3\sqrt{3} \]
Substituting \(a + \frac{1}{a} = \sqrt{3}\) into the equation:
\[ a^3 + \frac{1}{a^3} + 3\sqrt{3} = 3\sqrt{3} \]

### Step 5: Isolate \(a^3 + \frac{1}{a^3}\)
Now, we can isolate \(a^3 + \frac{1}{a^3}\):
\[ a^3 + \frac{1}{a^3} = 3\sqrt{3} - 3\sqrt{3} \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} = 0 \]

### Final Answer
Thus, the value of \(a^3 + \frac{1}{a^3}\) is:
\[ \boxed{0} \]