If `(a + 1/a)^2=3` then what is the value of
`a^3 + 1/(a^3) ? `

Options

A. `3 sqrt(3)`
B. `3`
C. `0`
D. `9`

Answer

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

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

### Step 1: Take the square root of both sides
Taking the square root of both sides gives us:

\[ a + \frac{1}{a} = \sqrt{3} \quad \text{or} \quad a + \frac{1}{a} = -\sqrt{3} \]

### Step 2: Use the identity for cubes
We want to find \( a^3 + \frac{1}{a^3} \). We can use the identity:

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

### Step 3: Substitute the value of \( a + \frac{1}{a} \)
Substituting \( a + \frac{1}{a} = \sqrt{3} \):

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

### Step 4: Calculate \( \left(\sqrt{3}\right)^3 \)
Calculating \( \left(\sqrt{3}\right)^3 \):

\[ \left(\sqrt{3}\right)^3 = 3\sqrt{3} \]

### Step 5: Substitute back into the equation
Now substituting back into our equation:

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

### Step 6: Simplify the expression
This simplifies to:

\[ a^3 + \frac{1}{a^3} = 0 \]

### Conclusion
Thus, the value of \( a^3 + \frac{1}{a^3} \) is:

\[ \boxed{0} \]