Well the first thing I did was check that they do intersect.
I did this by hand but here Desmos has done it.
ok so there are 2 points.
\(y=\sqrt{4-(x-4)^2}+3\\ y=2^{x-4}+3\\~\\ \sqrt{4-(x-4)^2}+3=2^{x-4}+3\\ \sqrt{4-(x-4)^2}=2^{x-4}\\ \sqrt{-x^2+8x-12}=2^{x-4}\\ 2^{4-x}\sqrt{-x^2+8x-12}=1\\ 2^{4-x}\sqrt{-x^2+8x-12}-1=0\\ \)
I don't know how to solve that ....
Here is what Wolfram|Alpha has to say about it.
http://www.wolframalpha.com/input/?i=sqrt%7B4-(x-4)%5E2%7D%2B3%3D2%5E(x-4)%2B3
That is it from me.