Arrange the following number from smallest to largest: $2^{1000}, 3^{750}, 5^{500}$.
Since all the numbers are positive, taking the 250th root of each number will not change their ordering. The resulting numbers are $2^{4}=16,3^{3}=27$, and $5^{2}=25$. These have the ordering $2^{4}<5^{2}<3^{3}$, so the ordering of the original numbers is $2^{1000}<5^{500}<3^{750}$.