How to arrange in order from smaller to biggest : $2^{120}$, $3^{72}$, $17^{30}$

Question

Comparison between exponents of different base and power.

Answer 1

Hint:

$$17^{30}>16^{30}=2^{4\cdot30}=2^{120}$$

Answer 2

$2^{120}=(2^{10})^{12}=1024^{12}>729^{12}=(3^6)^{12}=3^{72} \tag{1}$

In general, first take the divisors of lcm of powers and compare or first compare and then take the divisors of lcm of powers, whichever makes comparison easier.

I have use the technique mentioned in bold, while Roman83 has used the latter.

From $(1)$ and Roman83's answer, we get

$$17^{30} > 2^{120} > 3^{72}$$