Compare two powers of numbers without common divisor

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 $$3^{43}\gt 3^{42}=(3^7)^6\gt (2^{10})^6=2^{60}.$$

If we look at the powers of 3: 3, 9, 27, 81, 243, 729, 2187. 2187 looks pretty close to a power of 2: 2048. So let's start with that:

$$3^7 > 2^{11}$$

Take both to the 5th power:

$$3^{35} > 2^{55}$$

Obviously:

$$3^8 > 2^5$$

Multiplying those together: $3^{43} > 2^{60}$.