यदि `x+(1)/(x)=sqrt(3)` है तब `x^(3)+(1)/(x^(3))` का मान होगा -

Options

A. `2sqrt(3)`
B. `sqrt(3)(sqrt(3)-1)`
C. `3sqrt(3)`
D. 0

Answer

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

### Step 1: Square the given equation
We start with the equation:
\[ x + \frac{1}{x} = \sqrt{3} \]
Now, we square both sides:
\[ \left( x + \frac{1}{x} \right)^2 = (\sqrt{3})^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 3 \]

### Step 2: Rearrange to find \( x^2 + \frac{1}{x^2} \)
Now, we can rearrange the equation:
\[ x^2 + \frac{1}{x^2} = 3 - 2 = 1 \]

### Step 3: Use the identity for cubes
Next, we use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we have:
\[ x^3 + \frac{1}{x^3} = \left( \sqrt{3} \right)(1) - \sqrt{3} \]

### Step 4: Simplify the expression
Now, we simplify:
\[ x^3 + \frac{1}{x^3} = \sqrt{3} - \sqrt{3} = 0 \]

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