Every single online resource I've looked up on Akra-Bazzi method appears to skip over the same step: They say you have to solve for $p$ without explaining how. If you look up the various PDFs and webpages online, they all basically say "Here's the equation. Solve for $p$. It equals (whatever). Moving on now to the integral..." without explaining how they solved the equation.
If I had to figure it out I could use a binary search but I imagine there is a more mathematical way to get an exact answer.
It is not at all obvious to me how you're supposed to solve for p.
$$ a_1 b_1^p + a_2 b_2^p + a_3 b_3^p + \dots + a_k b_k^p = 1, $$
Where all $a_i$ terms are positive and all $b_i$ terms are fractional ($0 < b_i < 1$).
How can you solve for $p$ by hand?