$1 \cdot 44$ increasing sequence $2,3,5,6,7,10,11, \cdots$, includes all positive integers that are neither perfect squares nor perfect cubes. Try to find the 500th term of this sequence.
[Solution] It is easy to know that $[\sqrt{500}]=22, [\sqrt[3]{500}]=7, [\sqrt[6]{500}]=2$ (where $[x]$ represents the greatest integer not exceeding $x$).

Therefore, there are $22+7-2=27$ square numbers or cube numbers within the natural number 500. Now consider how many square numbers or cube numbers there are from $501 \sim 530$. By calculation, there is only one number $512=8^{3}$. Therefore,
$$a_{500}=500+27+1=528 \text {. }$$