If x=\frac{\sqrt{3}+1}{\sqrt{3}-1} and y=\frac{\sqrt{3}-1}{\sqrt{3}+1}, then what is the value of x^{3}-y^{3}?

Explanation

Rationalizing x: x = \frac{(\sqrt{3}+1)^2}{3-1} = 2+\sqrt{3}. Its conjugate is y = 2-\sqrt{3}. Thus, x-y = 2\sqrt{3} and xy=1. Using the identity x^3-y^3 = (x-y)((x-y)^2+3xy), we get 2\sqrt{3}((2\sqrt{3})^2+3(1)) = 2\sqrt{3}(12+3) = 30\sqrt{3}.

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