Is $202^{303}$ greater or $303^{202}$?

$$\array{808 &\gt& 9 \\ 101 \cdot 2^{3} &\gt& 3^{2} \\ 101^{3} \cdot 2^{3} &\gt& 101^{2} \cdot 3^{2} \\ \left((101\cdot 2)^{3}\right)^{101} &\gt& \left((101\cdot 3)^{2}\right)^{101} \\ 202^{303} &\gt& 303^{202}}$$

We have $a^b<b^a$ iff $b\log(a)<a\log(b)$ iff $\frac{\log(a)}a<\frac{\log(b)}b$. Consider the function: $$f:x\mapsto\frac{\log(x)}x:\mathbb R^+\to\mathbb R$$ Then: $$f'(x)=\frac{1-\log(x)}{x^2}$$ Hence $$f'(x)>0\quad\mathrm{iff}\quad x<e$$ Hence $f$ is decreasing for $x>e$ and this proves $f(303)<f(202)$, hence $303^{202}<202^{303}$.

We are comparing $202^{303}$ and $303^{202}$.

$202^{303}$ is equal to $202^{202}$ * $202^{101}$.

$303^{202}$ is equal to $(202 * 1.5)^{202}$ which is equal to $202^{202}$ * $1.5^{202}$

Now, we can divide out the $202^{202}$ from both sides which yields $202^{101}$ versus $1.5^{202}$. $1.5^{202}$ can be written as $2.25^{101}$ (squaring the inside, thus dividing the exponent by 2). Since $202^{101}$ > $2.25^{101}$, $202^{303}$ > $303^{202}$. No need for calculus!