Is there an operation that takes $a^b$ and $a^c$, and returns $a^{bc}$? I know that multiplying exponents of the same base will give you that base to the power of the sum of the exponents ($a^b \times a^c = a^{b+c}$), but is there anything that can be done with exponents that will give you some base to the power of the product of those exponents: $f(a^b, a^c) = a^{bc}$?

There is not.  Consider as one example  $$f(2^4, 2^4) = 2^{4\cdot 4} = 2^{16} =65536.$$  And you also want $$f(4^2, 4^2) = 4^{2\cdot 2} = 4^4 = 256.$$
But the left sides are equal (because $2^4=4^2$) while the right sides are not.  So the arguments of the function do not have enough information to produce the results you want.
