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

1 Answer

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#