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If $$x = 2 + \sqrt{3}$$, then the value of $$x^3 + x^{-3}$$ is
Since $$\frac{1}{x} = \frac{1}{2 + \sqrt{3}} = 2 - \sqrt{3}$$, we get $$x + \frac{1}{x} = 4$$. Then $$x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) = 64 - 12 = 52$$.
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