Question

If $n$ is a positive integer, then $(\sqrt{3}+1{)}^{2n}-(\sqrt{3}-1{)}^{2n}$ is :

• A. an irrational number
• B. an odd positive interger
• C. an even positive interger
• D. a rational number other than positive integers

Answer 1

The correct option is A an irrational number

$(\sqrt{3}+1{)}^{2n}-(\sqrt{3}-1{)}^{2n}={\left[{(\sqrt{3}+1)}^2\right]}^n-{\left[{(\sqrt{3}-1)}^2\right]}^n=(4+2\sqrt{3}{)}^n-(4-2\sqrt{3}{)}^n=2^n[{(2+\sqrt{3})}^n-{(2-\sqrt{3})}^n]=2^n\times 2[{}^n{C}_1{2}^{n-1}\sqrt{3}{+}^n{C}_3{.2}^{n-3}3\sqrt{3}+...]=2^{n+1}\sqrt{3}[{}^n{C}_1{2}^{n-1}{+}^n{C}_3{2}^{n-3}3+....]$
$=\sqrt{3}\times$Some integer $\therefore$ irrational number