Solution
In binomial expansion of $$\left(1+x\right)^n$$, if x is really close to 0, it could be written as: $$(1 + x)^n \approx 1 + nx$$.
So, $$1.001001001^{\frac{1}{3}}$$ can be written as $$1+\frac{1}{3}\times\ 0.001001001$$ and,
$$1.001001^{\frac{1}{3}}$$ can be written as $$1+\frac{1}{3}\times\ 0.001001$$.
Their difference comes out to be: $$\frac{1}{3}\left(0.001001001-0.001001\right)=\frac{0.000000001}{3}=\frac{1}{3}\times\ 10^{-9}=0.33\times\ 10^{-9}=3.3\times10^{-10}$$.
