How do we compare $8^{2700}$ and $3^{5500}$ without using the logarithm?

Question

I am trying to explain this to children with no knowledge of logarithm. We know that $8^{2700} = 2^{8100}$ and $3^{5500}$ is already in prime base. However, how do we know which one is larger in value?

I tried to consider the ratio of the two expressions, but I failed to continue with the argument. Any pointer will be appreciated.

Answer 1

We know that $$3^{5500}=9^{2750}>8^{2750}>8^{2700}$$ and we're done.

Answer 2

$$3^{5500}=(3^{55})^{100}, 8^{2700}=(8^{27})^{100}$$

So now, we have to compare $3^{55}$ and $8^{27}$. Notice that

$$8^{27}<9^{27}=3^{54}<3^{55}$$

So $3^{55}>8^{27}$, which will in turn mean that $3^{5500}>8^{2700}$.