Approximate a cube root with a binomial series

$\sqrt[3]{8+x}=2(1+x/8)^{1/3}=2[1+\tfrac13(x/8)+\tfrac{(1/3)(-2/3)}{2!}(x/8)^2+\cdots]=2+x/12-x^2/288+O(x^3)$. The printed MF27 validity condition is $|x/8|<1$, giving $-8<x<8$.

At $x=1/27$, $\sqrt[3]{217/27}\approx2+1/324-1/209952=420551/209952$. Since $\sqrt[3]{217}=3\sqrt[3]{217/27}$, $\sqrt[3]{217}\approx420551/69984$, in lowest terms.

The printed solution answers “better” because $1/217$ is closer to zero. This compares local truncation for $\sqrt[3]{8+x}$, but the stated substitution estimates $\sqrt[3]{8+1/217}=\sqrt[3]{1737/217}$, not $\sqrt[3]{217}$. The source supplies no transformation between these targets, so the claimed approximation of $\sqrt[3]{217}$ is not established as stated. A smaller expansion variable alone does not complete this comparison.