If `c+(1)/(c )=sqrt(3)`, then the value of `c^(3)+(1)/(c^(3))` is equal to

Options

A. `0`
B. `3sqrt(3)`
C. `(1)/(sqrt(3))`
D. `6sqrt(3)`

Answer

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

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

### Step 2: Expand the left-hand side using the identity
Using the identity \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), where \( a = c \) and \( b = \frac{1}{c} \):
\[ c^3 + \frac{1}{c^3} + 3 \left( c \cdot \frac{1}{c} \right) \left( c + \frac{1}{c} \right) = 3\sqrt{3} \]
Since \( c \cdot \frac{1}{c} = 1 \), we can simplify this to:
\[ c^3 + \frac{1}{c^3} + 3(c + \frac{1}{c}) = 3\sqrt{3} \]

### Step 3: Substitute the known value
We know that \( c + \frac{1}{c} = \sqrt{3} \). Substitute this into the equation:
\[ c^3 + \frac{1}{c^3} + 3\sqrt{3} = 3\sqrt{3} \]

### Step 4: Isolate \( c^3 + \frac{1}{c^3} \)
Now, we can isolate \( c^3 + \frac{1}{c^3} \):
\[ c^3 + \frac{1}{c^3} = 3\sqrt{3} - 3\sqrt{3} \]
This simplifies to:
\[ c^3 + \frac{1}{c^3} = 0 \]

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