The reaction rate R R\,R of a biochemical catalyst depends on its concentration ccc (in mol dm−3^{-3}−3) according to the model
R(c)=1+8c23,∣c∣<18 R(c) = \sqrt[3]{1 + 8c^2}, \quad |c| < \frac{1}{\sqrt{8}} R(c)=31+8c2,∣c∣<81Find, in ascending powers of ccc, the first three non-zero terms of the binomial series expansion of R(c)R(c)R(c), giving each coefficient as a simplified fraction.
By substituting c=14\displaystyle c = \frac{1}{4}c=41 into the expansion from part (a), find a rational approximation to k33k \sqrt[3]{3}k33, where k k\,k is a constant you must determine.