Question

Find ${\left(x+1\right)}^6+{\left(x-1\right)}^6$. Hence, or otherwise evaluate ${\left(\sqrt{2}+1\right)}^6+\sqrt{2}-1^6$.

Answer 1

The expression ${\left(x+1\right)}^6+{\left(x-1\right)}^6$ can be written as
$$\begin{gathered} (x+1{)}^6+(x-1{)}^6 \\ =2[{}^{6}C_0{x}^6+{}^{6}C_2{x}^4+{}^{6}C_4{x}^2+{}^{6}C_6{x}^0] \\ =2[x^6+15x^4+15x^2+1] \end{gathered}$$

By taking $x=\sqrt{2}$, we get:
$(\sqrt{2}+1{)}^6+(\sqrt{2}-1{)}^6=2\left[\right(\sqrt{2}{)}^6+15(\sqrt{2}{)}^4+15(\sqrt{2}{)}^2+1]$
$$\begin{gathered} =2[8+15\times 4+15\times 2+1] \\ =2\times (8+60+30+1) \\ =198 \end{gathered}$$