While solving an Algebra exercise I came across this expression $2^{a^b c}$ and simplified it in the following manner:
$$ 2^{a^b c} = 2^{(a^b)c} = (2^{a^b})^c = (2^a)^b)^c = (2^{ab})^c = 2^{abc}. $$
I have obviously done something wrong here. Not only because
$$2^{a^b c} = 2^{abc} \iff a^b c = abc,$$
but also because
$$2^{x^0 y} \overset{?}= 2^{x0y} = 1 \text{ and } 2^{x^0 y} = 2^y.$$
I can't for the life of me find out what I did wrong. All the steps I took seem justified. Can someone please point out where my mistake is?
$2^{a^bc}$ is $2^{a^bc}$, and that's all there is to it. Your assertion that $2^{a^b}=(2^a)^b$ is as false as a standalone statement as it is inside this more convoluted expression.