Homomorphic Encryption with Addition and Exponentiation
(In most generality..) you're looking for fully homomorphic encryption (FHE), which supports homomorphic addition and homomorphic multiplication. To compute a plaintext's exponentiation, you would run the repeating squaring algorithm (using homomorphic multiplication), to generate an (evaluated) homomorphic ciphertext containing the input-plaintext's exponentiation.

Craig Gentry gave the breakthrough construction of FHE in 2009. Since then, there's been a titanic volume of work improving the techniques, and multiple surveys are available by searching online.

An open-source implementation of modern FHE -- the library HElib by Shai Halevi and Victor Shoup -- can be found here: https://github.com/shaih/HElib