TITLE: Finding roots of equation
QUESTION [5 upvotes]: $$r\left(\frac 1p -1\right)- \left(\frac 1p\right)^\frac89 + \left(\frac 1p\right)^\frac19 =0$$
where $ r = 2^\frac89 - 2^\frac19$
How do I solve this without a computer?

REPLY [3 votes]: First of all, there no need to express as 1/p so
$(2^{\frac 8 9} - 2^\frac 1 9)(m - 1) - m^{\frac 8 9} + m^{\frac 1 9} = 0$
$(2^{\frac 8 9} - 2^\frac 1 9)(m - 1) - (m^{\frac 8 9} - m^{\frac 1 9}) = 0$
Oh look! The $(m^{\frac 8 9} - m^{\frac 1 9})$ looks like  $(2^{\frac 8 9} - 2^\frac 1 9)$!  What if $m = 2$?  Then $m - 1$ = 1 and we'd get an equality.  So $p = \frac 1 2$ is a solution.
But that was us seeing something.  Ho do we solve in general?
$(2^{\frac 8 9} - 2^\frac 1 9) = \frac{ (m^{\frac 8 9} - m^{\frac 1 9})}{m-1} $
$2^{\frac 1 9}(2^8 - 2) = \dfrac {m'^8 - m} {m'^9 - 1}$ for $m' = m^{\frac 1 9}$....
You know what? let's just say $p = \frac 1 2$ and call it a day.