If a+1/a=2 what is a^5+(1/a)^5?

1 Answer
Aug 14, 2017

If a+1/a=2 then a^5+(1/a)^5=color(red)(2)

Explanation:

If a+1/a=2
then (after multiplying everything on both sides by a
color(white)("XXX")a^2+1=2a
or, re-arranging
color(white)("XXX")a^2-2a+1=0
which can be factored as
color(white)("XXX")(a-1)(a-1)=0
implying that
color(white)("XXX")a=1

If a=1 then, a^5 = 1^5=1
and (1/a)^5=(1/1)^5=1

Therefore
color(white)("XXX")a^5+(1/a)^5=1+1=2