Which number is greater, $11^{11}$ or $9^{12}$? 
Which number is greater than $11^{11}$ or $9^{12}$?

My work so far:
$11^{11}=285311670611>9^{12}=282429536481$.
But to verify the validity of equality should be in the range of easily verifiable calculations.

This is a variant on Servaes's answer.  Note that $3^5=243=2\cdot11^2+1$.  Using a binomial expansion and some extremely crude upper bounds, we find
$$\begin{align}
3\cdot9^{12}
&=3^{25}\\
&=(2\cdot11^2+1)^5\\
&=32\cdot11^{10}+80\cdot11^8+80\cdot11^6+40\cdot11^4+10\cdot11^2+1\\
&\lt32\cdot11^{10}+80\cdot11^8+11^8+11^8+11^8+11^8\\
&\lt32\cdot11^{10}+121\cdot11^8\\
&=32\cdot11^{10}+11^{10}\\
&=3\cdot11^{11}
\end{align}$$
and thus $9^{12}\lt11^{11}$.
