Question

Factorise:

${\displaystyle \frac{8{\text{a}}^3}{27}-\frac{{\text{b}}^3}{8}}$

Answer 1

${\displaystyle \frac{8{\text{a}}^3}{27}-\frac{{\text{b}}^3}{8}}$

= ${\displaystyle {\left(\frac{2\text{a}}{3}\right)}^3-{\left(\frac{\text{b}}{2}\right)}^3}$

= ${\displaystyle (\frac{2\text{a}}{3}-\frac{\text{b}}{2})[{\left(\frac{2\text{a}}{3}\right)}^2+\frac{2\text{a}}{3}\times \frac{\text{b}}{2}+{\left(\frac{\text{b}}{2}\right)}^2]}$     ...[∵ a³ - b³ = (a - b)(a² + ab + b²)]

= ${\displaystyle (\frac{2\text{a}}{3}-\frac{\text{b}}{2})[\frac{{\text{4a}}^2}{9}+\frac{\text{ab}}{3}+\frac{{\text{b}}^2}{4}]}$