Compare two powers of numbers without common divisor
Which of the numbers $2^{60}$ and $3^{43}$ is greater? There is no common divisor and it must be done without a calculator.
We could also notice that
$3^{43} > 3^{40} = 9^{20} > 8^{20} = 2^{60}$.
Since $$3^7=2187\gt 1024=2^{10},$$ one has $$3^{43}\gt 3^{42}=(3^7)^6\gt (2^{10})^6=2^{60}.$$