If `a+1/a=sqrt3`, the value of `a^3+1/a^3` is :

Answer

To find the value of \( a^3 + \frac{1}{a^3} \) given that \( a + \frac{1}{a} = \sqrt{3} \), we can follow these steps:

### Step 1: Cube both sides of the equation
We start with the equation:
\[ a + \frac{1}{a} = \sqrt{3} \]
Now, we will cube both sides:
\[ \left(a + \frac{1}{a}\right)^3 = \left(\sqrt{3}\right)^3 \]

### Step 2: Expand the left side using the binomial theorem
Using the formula \( (x + y)^3 = x^3 + y^3 + 3xy(x + y) \), we can expand the left side:
\[ a^3 + \frac{1}{a^3} + 3\left(a \cdot \frac{1}{a}\right)\left(a + \frac{1}{a}\right) = 3\sqrt{3} \]
Since \( a \cdot \frac{1}{a} = 1 \), this simplifies to:
\[ a^3 + \frac{1}{a^3} + 3(a + \frac{1}{a}) = 3\sqrt{3} \]

### Step 3: Substitute \( a + \frac{1}{a} \) into the equation
We know that \( a + \frac{1}{a} = \sqrt{3} \), so substituting this into the equation gives:
\[ a^3 + \frac{1}{a^3} + 3\sqrt{3} = 3\sqrt{3} \]

### Step 4: 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:
\[ \boxed{0} \]