The infinite product √[3]10×√[3]√[3]10×√[3]√[3]√[3]10… evaluates to a real number. What is that number?

2022 AMC 12A Problem 8

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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?

\sqrt{10}

\sqrt[3]{100}

\sqrt[4]{1000}

10

10\sqrt[3]{10}

Write every factor as a power of

10

The exponents are

\dfrac13,\dfrac19,\dfrac1{27},\ldots,

a geometric series

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.