1. Which of the numbers is greater $2^{5^{4^{3}}}$ or $3^{4^{2^{5}}}$?
Solution:

$$
2^{5^{4^{3}}} \vee 3^{4^{2^{5}}} \Leftrightarrow 2^{5^{64}} \vee 3^{4^{32}} \Leftrightarrow 2^{5^{64}}>2^{4^{64}}=2^{4^{63} \cdot 4}=\left(2^{4}\right)^{4^{63}}=16^{4^{63}}>3^{4^{63}}>3^{4^{32}}
$$

Answer: $2^{5^{4^{3}}}>3^{4^{2^{5}}}$