Question

How do you use the binomial series to expand `(1-x^3)^(1/2)`?

Answer 1

You can't - the Binomial Theorem only applies to binomials raised to integer powers.

You can maybe recognize this more clearly if you rewrite it as a root:
`(1-x^3)^(1/2)=sqrt(1-x^3)`

There are no ways to "expand" a square root, so you really can't do anything unless you know what `x` is equal to.