2022 AMC 12A Problem 8

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.