Order the numbers $2^{300}, 10^{100}$, and $3^{200}$ from least to greatest, and prove that your ordering is correct.
We note that $2^{300}=\left(2^{3}\right)^{100}=8^{100}$ and $3^{200}=\left(3^{2}\right)^{100}=9^{100}$. Since $8^{100}<9^{100}<10^{100}$, the ordering is $2^{300}<3^{200}<10^{100}$.