[Exponential Inequalities]

Arrange the numbers in ascending order: $222^{2} ; 22^{22} ; 2^{222} ; 22^{2^{2}} ; 2^{22^{2}} ; 2^{2^{22}} ; 2^{2^{2^{2}}}$. Justify your answer.
First, let's consider the powers with base 2 and compare them: $2^{2^{2}}=2^{4}=16$, $16^{4}=2^{16}=2^{2^{2^{2}}}$, and $22^{2^{2}}=22^{4}<22^{22}<64^{37}=\left(2^{6}\right)^{37}=2^{222}$; $222^{2}<256^{2}=2^{16}=2^{2^{2^{2}}}$.

## Answer

$222^{2}<2^{2^{2^{2}}}<22^{2^{2}}<22^{22}<2^{222}<2^{22^{2}}<2^{2^{22}}$