Q:

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. 

A:

$$
\frac{11^{11}}{9^{11}}
=\left(\frac{2}{9}+1\right)^{11}
= \sum_{k=0}^{11} \binom{11}{k} \left(\frac{2}{9}\right)^k 
> \sum_{k=0}^5 \binom{11}{k} \left(\frac{2}{9}\right)^k = \frac{177665}{19683}
> 9
$$

