Let's determine the value of the product

$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$

when $n$ is infinitely large.
The given product can also be written as:

$$
z=a \cdot a^{\frac{1}{2}} \cdot a^{\frac{1}{4}} \cdot a^{\frac{1}{8}} \cdots a^{\frac{1}{2^{n}}}=a^{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{n}}}
$$

The exponent of $a$, when $n$ is infinitely large, is an infinite geometric series with a common ratio of $\frac{1}{2}$, and thus the exponent is:

$$
\frac{1}{1-\frac{1}{2}}=2
$$

therefore

$$
z=a^{2}
$$

(Jenö Silbermann, Nagyvárad.)