Question

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

Answer 1

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`