Which is bigger: $9^{10}$ or $10^{9}$?

Now find a way to use your calculator to compare $99^{100}$ and $100^{99}$.

Work out which is bigger out of $999^{1000}$ and $1000^{999}$.

NOTES AND BACKGROUND

This challenge calls for some experimentation with numbers and for some ingenuity in finding a solution. There are several possible methods of solution.

Optional extension challenge: If you solved the above, you might wish to consider which is bigger for the same sort of problem with a billion 9s and a billion 0s.