jmc

Let's try counting the number of perfect squares and cubes less than $441=21^2$. There are twenty perfect squares less than 441: $1^2,2^2,\dots ,20^2$. There are also seven perfect cubes less than 441: $1^3,2^3,\dots ,7^3$. So there would seem to be $20+7=27$ numbers less than 441 which are either perfect squares and perfect cubes.

But wait! $1=1^2=1^3$ is both a perfect square and a perfect cube, so we've accidentally counted it twice. Similarly, we've counted any sixth power less than 441 twice because any sixth power is both a square and a cube at the same time. Fortunately, the only other such one is $2^6=64$. Thus, there are $27-2=25$ numbers less than 441 that are perfect squares or perfect cubes. Also, since $20^2=400$ and $7^3=343$, then all 25 of these numbers are no more than 400. To compensate for these twenty-five numbers missing from the list, we need to add the next twenty-five numbers: 401, 402, $\dots$, 424, 425, none of which are perfect square or perfect cubes. Thus, the $400^{\text{th}}$ term is $\boxed{425}$.