Problem

(a) Find, in ascending powers of $x$, the first three non-zero terms of the binomial series expansion of

$$\sqrt[3]{1+4x^3}\qquad |x|<\frac{1}{\sqrt[3]{4}}$$

giving each coefficient as a simplified fraction.

(b) Use the expansion from part (a) with $x=\dfrac{1}{3}$ to find a rational approximation to $\sqrt[3]{31}$