Question

Evaluate $(\sqrt{3}-\sqrt{2}{)}^6+(\sqrt{3}+\sqrt{2}{)}^6$

• A. $678$
• B. $950$
• C. $970$
• D. $670$

Answer 1

The correct option is B $950$

Formula $(1+x{)}^n=k{}^{n}Co+{}^n{C}_{1}x+{}^n{C}_2{x}^2+....+{}^n{C}_k{x}^4+....+{}^n{C}_n{x}^n$ $=\sum _{k=0}^n$ ${}^n{C}_k{x}^n$

$(\sqrt{3}-\sqrt{2}{)}^6=(\sqrt{3}{)}^6-6\left(\sqrt{3}{)}^5\right(\sqrt{2})+15(\sqrt{3}{)}^4(\sqrt{2}{)}^2-\cdots$ $-6\left(\sqrt{3}\right)(\sqrt{2}{)}^5+(\sqrt{2}{)}^6$ ....... $\left(i\right)$

$(\sqrt{3}+\sqrt{2}{)}^6=(\sqrt{3}{)}^6+6\left(\sqrt{3}{)}^5\right(\sqrt{2})+15(\sqrt{3}{)}^4+(\sqrt{2}{)}^2+\cdots$ $+6\left(\sqrt{3}\right)(\sqrt{2}{)}^5+(\sqrt{2}{)}^6$ ...... $\left(ii\right)$

Adding eqn $\left(i\right)$ and $\left(ii\right),$ we get

$=1(\sqrt{3}{)}^6+2(15\sqrt{3}{)}^4({\sqrt{2}}^2+2\times 15(\sqrt{3}{)}^2(\sqrt{2}{)}^4+2(\sqrt{2}{)}^6$

$=2[3^3+15\times 3^2\times 2+15\times 3\times 2^2+2\times 2^3]$

$=2[9+270+196]$

$=950$