Question

Which is greater ${31}^{12}or{17}^{17}$

Answer 1

${17}^{17}$ is greater. We can use approximation method as follows:
Let us divide one by another:
$\frac{{17}^{17}}{{31}^{12}}\frac{(8\times 2{)}^{17}}{(15\times 2{)}^{12}}=\frac{8^{17}\times 2^5}{{15}^{12}}\frac{8^{17}\times 32}{{15}^{12}}\frac{8^{17}\times 2}{{15}^{11}}\frac{8^{17}\times 2}{(8^{11}\times 2^{11})}\frac{8^6\times 2}{2^{11}}=\frac{2^{19}}{2^{11}}=2^8.>1Hence,{17}^{17}>{31}^{12}.$
Actual values are ${17}^{17}=8.2724\times {10}^{20}and{31}^{12}is7.878\times {10}^{17}$