If (2 + \sqrt{3})^x + (2 - \sqrt{3})^x = 2, then what is (2 + \sqrt{3})^x - (2 - \sqrt{3})^x equal to?

Explanation

Let (2+\sqrt{3})^x = t. Since (2+\sqrt{3})(2-\sqrt{3}) = 1, (2-\sqrt{3})^x = \frac{1}{t}. The equation becomes t + \frac{1}{t} = 2, which gives t^2 - 2t + 1 = 0 \Rightarrow t = 1. Thus, (2+\sqrt{3})^x = 1, and (2-\sqrt{3})^x = 1. Their difference is 1 - 1 = 0.

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