Question

What is the formula for ${a^3} - {b^3}$?

Answer 1

Hint – In this question use the algebraic identity of ${\left( {a - b} \right)^3}$ which is ${a^3} - {b^3} - 3{a^2}b + 3a{b^2}$. Then take terms of this to a different side leaving behind only ${a^3} - {b^3}$. Simplification of the other side will help getting the right answer for ${a^3} - {b^3}$.

Complete step-by-step answer:
As we know that ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}$
$ \Rightarrow {a^3} - {b^3} = {\left( {a - b} \right)^3} + 3{a^2}b - 3a{b^2}$
$ \Rightarrow {a^3} - {b^3} = {\left( {a - b} \right)^3} + 3ab\left( {a - b} \right)$
Now take (a – b) common we have,
$ \Rightarrow {a^3} - {b^3} = \left( {a - b} \right)\left[ {{{\left( {a - b} \right)}^2} + 3ab} \right]$
Now as we know that ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ so use this property in above equation we have,
$ \Rightarrow {a^3} - {b^3} = \left( {a - b} \right)\left[ {{a^2} + {b^2} - 2ab + 3ab} \right]$
Now simplify it we have,
$ \Rightarrow {a^3} - {b^3} = \left( {a - b} \right)\left[ {{a^2} + {b^2} + ab} \right]$
So this is the required formula of ${a^3} - {b^3}$.

Note – The formula for ${(a - b)^3}$ can easily be derived by writing ${(a - b)^3} = (a - b){(a - b)^2}$. Then using the algebraic identity that ${(a - b)^2} = {a^2} + {b^2} - 2ab$, we can write ${(a - b)^3} = (a - b)({a^2} + {b^2} - 2ab)$, on simplification of this right hand side will help us getting the value of${(a - b)^3}$. So if somehow we find difficulty remembering this formula then the above steps can be used to derive it immediately.