$\begin{aligned}
& (\sqrt{3}+1)^{2 n}-(\sqrt{3}-1)^{2 n}=\left[(\sqrt{3}+1)^2\right]^n-\left[(\sqrt{3}-1)^2\right]^n=(4+2 \sqrt{3})^n-(4-2 \sqrt{3})^n \
& =2^n\left[(2+\sqrt{3})^n-(2-\sqrt{3})^n\right] \
& =2^n\left{\left[{ }^n C_0 2^n+{ }^n C_1 2^{n-1} \sqrt{3}+{ }^n C_2 2^{n-2} 3+\cdots \cdot\right]-\left[{ }^n C_0 2^n-{ }^n C_1 2^{n-1} \sqrt{3}+{ }^n C_2 2^{n-2} 3-\cdots \cdot\right]\right} \
& =2^{n+1}\left[{ }^n C_1 2^{n-1} \sqrt{3}+{ }^n C_3 2^{n-3} 3 \sqrt{3}+\cdots \cdot\right]=2^{n+1} \sqrt{3} \text { (some integer) }
\end{aligned}$ Which is irrational