Question 27
Consider a sequence $x_1,$ $x_2,$ $x_3,$ $\ldots,$ defined by $$
\begin{aligned} x_1&=\sqrt[3]{3},\\ x_2&=(\sqrt[3]{3})^{\sqrt[3]{3}}. \end{aligned}
$$ and in general $$
x_n=(x_{n-1})^{\sqrt[3]{3}}
$$ for $n\gt1.$ What is the smallest value of $n$ for which $x_n$ is an integer?
\begin{aligned} x_1&=\sqrt[3]{3},\\ x_2&=(\sqrt[3]{3})^{\sqrt[3]{3}}. \end{aligned}
$$ and in general $$
x_n=(x_{n-1})^{\sqrt[3]{3}}
$$ for $n\gt1.$ What is the smallest value of $n$ for which $x_n$ is an integer?