AMC 12A 2022 Problem 8

The infinite product
\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \cdots

evaluates to a real number. What is that number?

Answer Choices
A. \sqrt{10}
B. \sqrt[3]{100}
C. \sqrt[4]{1000}
D. 10
E. 10 \sqrt[3]{10}