Which of the following numbers is greater? Which of the following numbers is greater?
Without using a calculator and logarithm.
$$7^{55} ,5^{72}$$
My try
$$A=\frac{7^{55} }{5^{55}×5^{17}}=\frac{ 7^{55}}{5^{55}}×\frac{1}{5^{17}}= \left(\frac{7}{5}\right)^{55} \left(\frac{1}{5}\right)^{17}$$
What now?

Consider that
$$
\frac{7^{55}}{5^{72}} = 7\left(\frac{7^3}{5^4}\right)^{18}
$$
Now, hand-calculation is easy enough on the bracketed term: $7^3=7\times49=343$ and $5^4=25^2=625$. As such, the bracketed term is just a little larger than $1/2$, and certainly less than $1/\!\sqrt{2}$. Therefore,
$$
7\left(\frac{7^3}{5^4}\right)^{18}<7\left(\frac12\right)^9 = \frac{7}{512}<1
$$
