1984 AHSME Problem 11

A calculator has a key that replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let y be the final result when one starts with an entry x\neq 0 and alternately squares and reciprocates n times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then y equals

\mathrm{(A) \ }x^{((-2)^n)} \qquad \mathrm{(B) \ }x^{2n} \qquad \mathrm{(C) \ } x^{-2n} \qquad \mathrm{(D) \ }x^{-(2^n)} \qquad \mathrm{(E) \ } x^{((-1)^n2n)}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.