We've started with $x^2$, saying that there are $\sqrt{10^9}$ numbers that are not $\in 10^9$
i'm thinking that if we add $\sqrt[3]{10^9}$ and $\sqrt[5]{10^9}$ to $\sqrt{10^9}$ and subtract them from $10^9$, we would have gone too far and excluded too many numbers due to repeating numbers such as $2^2$,$2^3$ and $2^5$, for example.
does it even make sense to do so? aren't all numbers in $\sqrt[3]{10^9}$ and $\sqrt[5]{10^9}$ $< 10^9$ already in $\sqrt{10^9}$ ?