If `x = 2 + sqrt(3)` then the value of `x^(3) + x^(-3)` is :

Options

A. 52
B. `-52`
C. `-52sqrt(3)`
D. `52sqrt(3)`

Answer

To find the value of \( x^3 + x^{-3} \) given that \( x = 2 + \sqrt{3} \), we can follow these steps:

### Step 1: Calculate \( x + x^{-1} \)

First, we need to find \( x^{-1} \):

\[ x^{-1} = \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \]

To simplify \( x^{-1} \), we can multiply the numerator and denominator by the conjugate of the denominator:

\[ x^{-1} = \frac{1 \cdot (2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} \]

Now, we can calculate \( x + x^{-1} \):

\[ x + x^{-1} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \]

### Step 2: Use the identity for \( x^3 + x^{-3} \)

We can use the identity:

\[ x^3 + x^{-3} = (x + x^{-1})^3 - 3(x + x^{-1}) \]

Substituting \( x + x^{-1} = 4 \):

\[ x^3 + x^{-3} = 4^3 - 3 \cdot 4 \]

### Step 3: Calculate \( 4^3 \) and \( 3 \cdot 4 \)

Calculating \( 4^3 \):

\[ 4^3 = 64 \]

Calculating \( 3 \cdot 4 \):

\[ 3 \cdot 4 = 12 \]

### Step 4: Substitute back into the equation

Now substitute these values back into the equation:

\[ x^3 + x^{-3} = 64 - 12 = 52 \]

### Final Answer

Thus, the value of \( x^3 + x^{-3} \) is:

\[ \boxed{52} \]