If $x$ and $y$ are positive integers with $x y=6$, the sum of all of the possible values of $\frac{2^{x+y}}{2^{x-y}}$ is
(A) 4180
(B) 4160
(C) 4164
(D) 4176
(E) 4128
Using exponent laws, the expression $\frac{2^{x+y}}{2^{x-y}}=2^{(x+y)-(x-y)}=2^{2 y}$.

Since $x$ and $y$ are positive integers with $x y=6$, then the possible values of $y$ are the positive divisors of 6 , namely $1,2,3$, or 6 . (These correspond to $x=6,3,2,1$.)

The corresponding values of $2^{2 y}$ are $2^{2}=4,2^{4}=16,2^{6}=64$, and $2^{12}=4096$.

Therefore, the sum of the possible values of $\frac{2^{x+y}}{2^{x-y}}$ is $4+16+64+4096=4180$.

ANSWER: (A)