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

Question

The infinite product $\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \ldots$ is

Professor PC Thomas Classes · Answer

Exponents aggregate as an infinite geometric series: $\frac{1}{3} + \frac{1}{9} + \dots = \frac{1}{2}$. Result is $10^{1/2} = \sqrt{10}$.