Recently while coming up with an example for a paper I'm writing I find myself wanting something to say about how 'awful' the first positive root of the equation $ 4\cdot2^{2p}\cdot3^p-4\cdot3^{2p}-7\cdot2^{3p}+8\cdot2^{2p}+8\cdot3^p-4=0 $ is. Numerically I know it's about 1.576, but I suspect that it's no only irrational but cannot be solved for using elementary functions, e.g. $\log_2(3+\sqrt{5})$.
I've not much / any background in number theory of things like this and am kind of stumped. Are there any simple arguments in this direction?
Note: This is the simplest of the awful equations I could come up with. I'm also aware that it has a root at $p=2$