jmc

We want to maximize $$|(3+4i)z^3-z^5|=|z^3||3+4i-z^2|=|z{|}^3|3+4i-z^2|=8|3+4i-z^2|.$$In other words, we want to maximize the distance between $3+4i$ and $z^2.$

Since $|z|=2,$ the set of complex numbers of the form $z^2$ lie on a circle with radius $|z{|}^2=4.$ The distance between $3+4i$ and $z^2$ is maximized when $z^2$ lies on the line passing through the origin and the $3+4i.$ (This line intersects the circle at two points, so we take the one that is farther from $3+4i.$)



For this number, the distance between $3+4i$ and $z^2$ is $4+5=9,$ so the maximum value of $8|3+4i-z^2|$ is $8\cdot 9=\boxed{72}.$