When $\sqrt[3]{2700}$ is simplified, the result is $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible.  What is $a+b$?
We have $$\sqrt[3]{2700} = \sqrt[3]{27}\times \sqrt[3]{100} = \sqrt[3]{3^3}\times \sqrt[3]{100} = 3\sqrt[3]{100}.$$  Since the prime factorization of 100 is $2^2\cdot5^2$, we cannot simplify $\sqrt[3]{100}$ any further.  Therefore, we have $a+b = \boxed{103}$.