If `(x + (1)/(x)) ^(2)= 3,` then the value of `x ^(3) + (1)/(x ^(3))` is

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

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

### Step 1: Simplify the given equation
Start with the equation:
\[ (x + \frac{1}{x})^2 = 3 \]
Taking the square root of both sides gives:
\[ x + \frac{1}{x} = \sqrt{3} \quad \text{or} \quad x + \frac{1}{x} = -\sqrt{3} \]

### Step 2: Use the identity for cubes
We can use the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x}) \]
Let \( k = x + \frac{1}{x} \). Then we can express \( x^3 + \frac{1}{x^3} \) as:
\[ x^3 + \frac{1}{x^3} = k^3 - 3k \]

### Step 3: Substitute the value of \( k \)
Using \( k = \sqrt{3} \):
\[ x^3 + \frac{1}{x^3} = (\sqrt{3})^3 - 3(\sqrt{3}) \]
Calculating \( (\sqrt{3})^3 \):
\[ (\sqrt{3})^3 = 3\sqrt{3} \]
Now substitute this back into the equation:
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} \]

### Step 4: Simplify the expression
This simplifies to:
\[ x^3 + \frac{1}{x^3} = 0 \]

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