<s>[INST] 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.
 [/INST] Just for the heck of it, here's another approach using the fact that
$$9^3=6\times11^2+3\qquad\text{ and }\qquad 6^4<11^3-3\times11.$$
It is by no means the slickest way;
\begin{eqnarray*}
9^{12}&=& (6\times11^2+3)^4=3^4\times(2\times11^2+1)^4\\
      &=& 3^4\times(2^4\times11^8+4\times2^3\times11^6+6\times2^2\times11^4+4\times2\times11^2+1)\\
&=&6^4\times(11^8+2\times11^6+\tfrac{3}{2}\times11^4+\tfrac{1}{2}\times11^2+\tfrac{1}{16})\\
&<&(11^3-3\times11)\times(11^8+2\times11^6+\tfrac{3}{2}\times11^4+\tfrac{1}{2}\times11^2+\tfrac{1}{16})\\
&=&11^{11}+2\times11^9+\tfrac{3}{2}\times11^7+\tfrac{1}{2}\times11^5+\tfrac{1}{16}\times11^3\\
&\ &\quad\ \ \ -3\times11^9-\ 6\times11^7-\tfrac{9}{2}\times11^5-\ \tfrac{3}{2}\times11^3-\tfrac{3}{2}\times11
\end{eqnarray*}
The alignment of the last expression shows that it is smaller than $11^{11}$, as desired.
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