List the numbers in order

Question

How would I list these numbers in order without using a calculator? Thank you

List these numbers in increasing order: $2^{800}$, $3^{600}$, $5^{400}$, $6^{200}$

Answer 1

Note that $$2^{800} = 16^{200}$$ $$3^{600} = 27^{200}$$ $$5^{400} = 25^{200}$$

so the order is: $6^{200}, 2^{800}, 5^{400}, 3^{600}$.

Answer 2

It is sufficient to compare among

$$2^4=16,3^3=27,5^2=25,6^1=6$$

as for positive real numbers $a,b>1$ and $n>0$ if $a>b\implies a^n>b^n$

here $n=200$