[tex]\dfrac{a+1}{a}=5\\\\\dfrac{a}{a}+\dfrac{1}{a}=5\\\\1+\dfrac{1}{a}=5\qquad\text{subtract 1 from both sides}\\\\\dfrac{1}{a}=4\\\\\left(\dfrac{1}{a}\right)^4=4^4\\\\\dfrac{1}{a^4}=256\qquad(*)\\-----------------------\\\\\dfrac{a^4+1}{a^4}=\dfrac{a^4}{a^4}+\dfrac{1}{a^4}=1+\underbrace{\dfrac{1}{a^4}}_{(*)}=1+256=257[/tex]
