If ab(a + b) = 1, then what is the value of $$\frac{1}{{{a^3}{b^3}}} - {a^3} - {b^3}?$$

Solution (By Examveda Team)

$$\eqalign{ & \because \,ab\left( {a + b} \right) = 1 \cr & \Rightarrow a + b = \frac{1}{{ab}}\,.......\,\left( {\text{i}} \right) \cr & {\text{On cubing both sides}} \cr & \Rightarrow {a^3} + {b^3} + 3ab\left( {a + b} \right) = \frac{1}{{{a^3}{b^3}}} \cr & \Rightarrow {a^3} + {b^3} + 3ab \times \frac{1}{{ab}} = \frac{1}{{{a^3}{b^3}}}\left( {{\text{from equation }}\left( {\text{i}} \right)} \right) \cr & \Rightarrow \frac{1}{{{a^3}{b^3}}} - {a^3} - {b^3} = 3 \cr} $$