Find the value
\[ x = 2 + \sqrt{3} \]
\[ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} = 2 – \sqrt{3} \]
\[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 – \sqrt{3}) = 4 \]
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 – 3(x + \frac{1}{x}) \]
\[ = 4^3 – 3(4) \]
\[ = 64 – 12 = 52 \]