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

Options

A. `-52`
B. `-30sqrt(3)`
C. `30sqrt(3)`
D. 52

Answer

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

### Step 1: Calculate \( x^3 \)

First, we need to find \( x^3 \). We can use the binomial expansion or directly calculate it.

\[ x = 2 + \sqrt{3} \]

Calculating \( x^2 \):

\[ x^2 = (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

Now, we can calculate \( x^3 \):

\[ x^3 = x \cdot x^2 = (2 + \sqrt{3})(7 + 4\sqrt{3}) \]

Using the distributive property:

\[ x^3 = 2 \cdot 7 + 2 \cdot 4\sqrt{3} + \sqrt{3} \cdot 7 + \sqrt{3} \cdot 4\sqrt{3} \]
\[ = 14 + 8\sqrt{3} + 7\sqrt{3} + 4 \cdot 3 \]
\[ = 14 + 8\sqrt{3} + 7\sqrt{3} + 12 \]
\[ = 26 + 15\sqrt{3} \]

### Step 2: Calculate \( x^{-1} \)

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

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

To rationalize the denominator, multiply by the conjugate:

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

### Step 3: Calculate \( x^{-3} \)

Now we can find \( x^{-3} \):

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

Using the binomial expansion:

\[ (2 - \sqrt{3})^3 = 2^3 - 3 \cdot 2^2 \cdot \sqrt{3} + 3 \cdot 2 \cdot (\sqrt{3})^2 - (\sqrt{3})^3 \]
\[ = 8 - 12\sqrt{3} + 3 \cdot 2 \cdot 3 - 3\sqrt{3} \]
\[ = 8 - 12\sqrt{3} + 18 - 3\sqrt{3} \]
\[ = 26 - 15\sqrt{3} \]

### Step 4: Calculate \( x^3 - x^{-3} \)

Now we can find \( x^3 - x^{-3} \):

\[ x^3 - x^{-3} = (26 + 15\sqrt{3}) - (26 - 15\sqrt{3}) \]
\[ = 26 + 15\sqrt{3} - 26 + 15\sqrt{3} \]
\[ = 30\sqrt{3} \]

### Final Answer

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

\[ \boxed{30\sqrt{3}} \]