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?

Observe that $7^2/5^2 = 49/25 < 2$.  Cubing both sides, $(7^2/5^2)^3 < 8$.  Since $7/5^3 = 7/125 < 1/8$ we have $7^7/5^9 < 8 \times 1/8 = 1$.
Then $7^{55}/5^{72} = 1/7 \times (7^7/5^9)^8 < 1/7  < 1$ and so $7^{55} < 5^{72}$.
