Question

Find the value of
${(x+\sqrt{2})}^4+{(x-\sqrt{2})}^4$

Answer 1

${(x+\sqrt{2})}^4+{(x-\sqrt{2})}^4$

$=\{x^4+4x^3(\sqrt{2})+6x^2(\sqrt{2}{)}^2+4x(\sqrt{2}{)}^3+(\sqrt{2}{)}^4\}+\{x^4-4x^3\left(\sqrt{2}\right)+6x^2(\sqrt{2}{)}^2-4x(\sqrt{2}{)}^3+\left(\sqrt{2}{)}^4\right\}$

$=2\{x^4+6x^2(\sqrt{2}{)}^2+4\left(\sqrt{2}{)}^4\right\}=2(x^4+12x^2+4)$