2022 AMC 12A Problem 8

Problem 8 of 25EasierAlgebra

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?

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Solution

The

k

th factor is

10

raised to the

k

-fold cube root, namely

10^{\frac{1}{3^k}}.

The product is

10

raised to

\begin{aligned} &\frac13+\frac19+\frac1{27}+\cdots=\frac{\frac{1}{3}}{1-\frac{1}{3}} \\ &=\frac12. \end{aligned}

So the value is

10^{\frac{1}{2}}=\sqrt{10}.

Thus, the correct answer is A.