1985 AHSME Problem 27

Consider a sequence x_1, x_2, x_3, \cdots defined by

x_1=\sqrt[3]{3}

x_2=(\sqrt[3]{3})^{\sqrt[3]{3}}

and in general

x_n=(x_{n-1})^{\sqrt[3]{3}} for n>1.

What is the smallest value of n for which x_n is an integer?

\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27

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