Mathematics - Binomial Theorem Question with Solution | TestHub

$\begin{array}{rl}& (\sqrt{3}+1{)}^{2n}-(\sqrt{3}-1{)}^{2n}={\left[(\sqrt{3}+1{)}^2\right]}^n-{\left[(\sqrt{3}-1{)}^2\right]}^n=(4+2\sqrt{3}{)}^n-(4-2\sqrt{3}{)}^n\\ & =2^n\left[(2+\sqrt{3}{)}^n-(2-\sqrt{3}{)}^n\right]\\ & =2^n\left\{\left[{}^n{C}_0{2}^n+{}^n{C}_1{2}^{n-1}\sqrt{3}+{}^n{C}_2{2}^{n-2}3+\cdots \cdot \right]-\left[{}^n{C}_0{2}^n-{}^n{C}_1{2}^{n-1}\sqrt{3}+{}^n{C}_2{2}^{n-2}3-\cdots \cdot \right]\right\}\\ & =2^{n+1}\left[{}^n{C}_1{2}^{n-1}\sqrt{3}+{}^n{C}_3{2}^{n-3}3\sqrt{3}+\cdots \cdot \right]=2^{n+1}\sqrt{3}\text{ (some integer) }\end{array}$ Which is irrational