Question

Find value of $(a^2+\sqrt{a^2-1}{)}^4+(a^2-\sqrt{a^2-1}{)}^4$ if a=$\sqrt{5}$.

Answer 1

$(a^2+\sqrt{a^2-1}{)}^4+(a^2-\sqrt{a^2-1}{)}^4$

putting $a=\sqrt{5}$.

$({\sqrt{5}}^2+\sqrt{{\sqrt{5}}^2-1}{)}^4+({\sqrt{5}}^2-\sqrt{{\sqrt{5}}^2-1}{)}^4=(5+\sqrt{5-1}{)}^4+(5-\sqrt{5-1}{)}^4$

$(5+\sqrt{4}{)}^4+(5-\sqrt{4}{)}^4=(5+2{)}^4+(5-2{)}^4=7^4+3^4=2482$