Which is larger,

$$
3^{3^{3^{3}}} \text { or } 2^{2^{2^{2^{2}}}} ?
$$

Remark. Note that $3^{3^{3}}$ means $3^{\left(3^{3}\right)}$, not $\left(3^{3}\right)^{3}$ and so on.
Starting to evaluate, the first expression becomes

$$
A=3^{3^{3^{3}}}=3^{3^{27}} .
$$

The second expression is

$$
B=2^{2^{2^{2^{2}}}}=2^{2^{2^{4}}}=2^{2^{16}} .
$$

It is now obvious that

$$
A=3^{3^{27}}>3^{3^{16}}>3^{2^{16}}>2^{2^{16}}=B
$$

so the first expression is the larger one.

![](https://cdn.mathpix.com/cropped/2024_04_17_08a0d1a184d1cdb14992g-1.jpg?height=222&width=225&top_left_y=301&top_left_x=1470)