What is the smaller number, $5^{2002}$ or $3^{2002}+4^{2002}$?
As $5^{2}=3^{2}+4^{2}$, we have $5^{2002}=\left(3^{2}+4^{2}\right)^{1001}$. We know that, for $a>0$ and $b>0$,

$$
(a+b)^{1001}>a^{1001}+b^{1001}
$$

Thus, $5^{2002}>3^{2002}+4^{2002}$.