Choose a valid substitution and assess the resulting estimate

(i) Expand $\sqrt[3]{\dfrac{x+2}{x-1}}$ in ascending powers of x, up to and including the term in $x^2$.

(ii) Find the range of values of x for which the expansion is valid.

(iii) By putting $x=8/11$ into the expansion found in (i), find an approximate value of $\sqrt[3]5$ correct to 4 decimal places.

(iv) Two students suggested alternative substitutions to find an approximate value of $\sqrt[3]5$.

Brian	Clare
$x=3$	$x=-1/3$

Without performing any calculations, identify the student who made an unacceptable estimate. Explain your answer.

(v) Use the substitution suggested by the other student to find another estimate for $\sqrt[3]5$ correct to 4 decimal places. Explain why this new approximation is better.