jmc

We see that it's easier to count the number of perfect squares, perfect cubes and perfect fifth powers less than $33^2=1089$. We see there are 32 perfect squares less than 1089, which are $1^2$, $2^2$, $\dots$, $32^2$ and then there are 10 perfect cubes which are $1^3$, $\dots$, $10^3$. There are 4 perfect fifth powers less than 1089 which are $1^5$, $\dots$, $4^5$. Then notice there are 3 numbers that are both perfect squares and perfect cubes which are 1, $2^6=64$ and $3^6=729$. There are also 2 numbers that are both perfect squares and perfect fifth powers which are $1^{10}=1$ and $2^{10}=1024$. The only number which is both a perfect cube and a perfect fifth power is $1^{15}=1$. The only number which is a perfect square, perfect cube, and perfect fifth power all at the same time is $1^{30}=1$. So within the first 1089 positive integers we need to get rid of $32+10+4-3-2-1+1=41$ integers which means the $1000^{\text{th}}$ term is $1000+41=\boxed{1041}$.