### Instruction: How many positive whole numbers have cube roots that are less than $10$? For example, $20$ would count since $\sqrt[3]{20}<10.$.### Response: I want to find the largest positive whole number whose cube root is still less than $10$.
If I cube both sides of the inequality $\sqrt[3]{n}<10$, I get $n<10^3=1000$.
So, any positive whole number less than $1000$ will have a cube root less than $10$.
There are $999$ such numbers, starting from $1$ and ending with $999$.
Therefore, the answer is $999$.