Let $a$ be the largest positive integer so that $a^{3}$ is less than 999.

Let $b$ be the smallest positive integer so that $b^{5}$ is greater than 99 .

What is the value of $a-b$ ?
Since $9^{3}=729$ and $10^{3}=1000$, then the largest positive integer $a$ with $a^{3}99$ is $b=3$.

Therefore, $a-b=6$.